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Evokation
  
 
Index
  

 

NUCLEAR FAMILY 19769

 

 

 

 

 

 

 

 

THE

MAGICALALPHABET

 

..................

 

 

-
-
-
-
-
THE RAINBOW LIGHT
-
-
-
T
=
2
-
3
THE
33
15
6
R
=
9
-
7
RAINBOW
82
37
1
L
=
3
-
5
LIGHT
56
29
2
-
-
14
-
15
THE RAINBOW LIGHT
171
81
9
-
-
1+4
-
1+5
-
1+7+1
8+1
-
Q
-
5
-
6
THE RAINBOW LIGHT
9
9
9

 

 

 

 

 

 

 

THIS IS THE SCENE OF THE SCENE UNSEEN

THE UNSEEN SEEN OF THE SCENE UNSEEN THIS IS THE SCENE

 

 

3
THE
33
15
6
4
MIND
40
22
4
2
OF
21
12
3
9
HUMANKIND
95
41
5
18
First Total
189
90
18
1+8
Add to Reduce
1+8+9
9+0
1+8
9
Second Total
18
9
9
-
Reduce to Deduce
1+8
-
-
9
Essence of Number
9
9
9

 

 

THE DIVINE COMEDY

OF

DANTE ALIGHIERI (1265-1321)

THE FLORENTINE

CANTICA I

HELL

(L'INFERNO)

INTRODUCTION

Page 9

"Midway this way of life we're bound upon

I woke to find myself in a dark wood,

Where the right road was wholly lost and gone."

 

 

M
=
4
-
6
MIDWAY
75
30
3
T
=
2
-
4
THIS
56
20
2
W
=
5
-
3
WAY
49
13
4
O
=
6
-
2
OF
21
12
3
L
=
3
-
4
LIFE
32
23
5
W
=
5
-
4
WE'RE
51
24
6
B
=
2
-
5
BOUND
56
20
2
U
=
3
-
4
UPON
66
21
3
-
-
30
-
32
-
406
163
28
-
-
-
-
-
-
-
-
-
I
=
9
-
1
I
9
9
9
W
=
5
-
4
WOKE
54
18
9
T
=
2
-
2
TO
35
8
8
F
=
6
-
4
FIND
33
24
6
M
=
4
-
6
MYSELF
80
26
8
I
=
9
-
2
IN
23
14
5
A
=
1
-
1
A
1
1
1
D
=
4
-
4
DARK
34
16
7
W
=
5
-
4
WOOD
57
21
3
-
-
45
-
28
-
326
137
56
-
-
-
-
-
-
-
-
-
W
=
5
-
5
WHERE
59
32
5
T
=
2
-
3
THE
33
15
6
R
=
9
-
5
RIGHT
62
35
8
R
=
9
-
4
ROAD
38
20
2
W
=
5
-
3
WAS
43
7
7
W
=
5
-
6
WHOLLY
95
32
5
L
=
3
-
4
LOST
66
12
3
A
=
1
-
3
AND
19
10
1
G
=
7
-
4
GONE
41
23
5
-
-
46
-
37
-
456
186
42
-
-
-
-
-
-
-
-
-
-
-
121
-
97
First Total
1188
486
126
-
-
1+2+1
-
9+7
Add to Reduce
1+1+8+8
4+8+6
1+2+6
Q
-
4
-
16
Second Total
18
18
9
-
-
-
-
1+6
Reduce to Deduce
1+8
1+8
-
-
-
4
-
7
Essence of Number
9
9
9

 

 

THE DIVINE COMEDY

OF

DANTE ALIGHIERI (1265-1321)

THE FLORENTINE

CANTICA I

HELL

(L'INFERNO)

INTRODUCTION

Page 9

"Power failed high fantasy here; yet, swift to move

Even as a wheel moves equal, free from jars,

Already my heart and will were wheeled by love,

The Love that moves the sun and other stars."

 

P
=
7
-
5
POWER
77
32
5
F
=
6
-
6
FAILED
37
28
1
H
=
8
-
4
HIGH
32
32
5
F
=
6
-
7
FANTASY
86
23
5
H
=
8
-
4
HERE
36
27
9
Y
=
7
-
3
YET
50
14
5
S
=
1
-
5
SWIFT
77
23
5
T
=
2
-
2
TO
35
8
8
M
=
4
-
4
MOVE
55
19
1
-
-
49
-
40
First Total
485
206
44
-
-
4+9
-
4+0
Add to Reduce
4+8+5
2+0+6
4+4
Q
-
13
-
4
Second Total
17
8
8
-
-
1+3
-
-
Reduce to Deduce
1+7
-
-
-
-
4
-
4
Essence of Number
8
8
8

 

 

E
=
5
-
4
EVEN
46
19
1
A
=
1
-
2
AS
20
2
2
A
=
1
-
1
A
1
1
1
W
=
5
-
5
WHEEL
53
26
8
M
=
4
-
5
MOVES
74
20
2
E
=
5
-
5
EQUAL
56
20
2
F
=
6
-
4
FREE
34
25
7
F
=
6
-
4
FROM
52
25
7
J
=
1
-
4
JARS
48
12
3
-
-
34
-
34
First Total
384
150
33
-
-
3+4
-
3+4
Add to Reduce
3+8+4
1+5+0
3+3
Q
-
7
-
7
Second Total
15
6
6
-
-
-
-
-
Reduce to Deduce
1+5
-
-
-
-
7
-
7
Essence of Number
6
6
6

 

 

A
=
1
-
7
ALREADY
66
30
3
M
=
4
-
2
MY
38
11
2
H
=
8
-
5
HEART
52
25
7
A
=
1
-
3
AND
19
10
1
W
=
5
-
4
WILL
56
20
2
W
=
5
-
4
WERE
51
24
7
W
=
5
-
7
WHEELED
62
35
8
B
=
2
-
2
BY
27
9
9
L
=
3
-
4
LOVE
54
18
9
-
-
34
-
38
First Total
425
182
47
-
-
3+4
-
3+8
Add to Reduce
4+2+5
1+8+2
4+7
Q
-
7
-
11
Second Total
11
11
11
-
-
-
-
1+1
Reduce to Deduce
1+1
1+1
1+1
-
-
7
-
2
Essence of Number
2
2
2

 

 

T
=
2
-
3
THE
33
15
6
L
=
3
-
4
LOVE
54
18
9
T
=
2
-
4
THAT
49
13
4
M
=
4
-
5
MOVES
74
20
2
T
=
2
-
3
THE
33
15
6
S
=
1
-
3
SUN
54
9
9
A
=
1
-
3
AND
19
10
1
O
=
6
-
5
OTHER
66
30
3
S
=
1
-
5
STARS
77
14
5
-
-
22
-
35
First Total
459
144
45
-
-
2+2
-
3+5
Add to Reduce
4+5+9
1+4+4
4+5
Q
-
4
-
8
Second Total
18
9
9
-
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
4
-
8
Essence of Number
9
9
9

 

 

3
THE
33
15
6
4
HOLY
60
24
6
7
MESSAGE
-
-
-
2
M+E
18
9
9
1
S
19
10
1
3
S+A+G
27
18
9
1
E
5
5
5
7
MESSAGE
69
42
24
14
Add
162
81
36
1+4
Reduce
1+6+2
8+1
3+6
5
Deduce
9
9
9

 

 

THE

FAR YONDER SCRIBE

AND OFT TIMES SHADOWED SUBSTANCES WATCHED IN FINE AMAZE

THE

ZED ALIZ ZED

IN SWIFT REPEAT SCATTER STAR DUST AMONGST THE LETTERS OF THEIR PROGRESS

AT THE THROW OF THE NINTH NUMBER WHEN IN CONJUNCTION SET

THE

FAR YONDER SCRIBE

MADE RECORD OF THEIR FALL

 

 

-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
A
=
1
-
5
ADDED
18
18
9
-
-
-
-
-
-
-
-
-
9
T
=
2
-
2
TO
35
8
8
-
-
-
-
-
-
-
-
8
-
A
=
1
-
3
ALL
25
7
7
-
-
-
-
-
-
-
7
-
-
M
=
4
-
5
MINUS
76
22
4
-
-
-
-
4
-
-
-
-
-
N
=
5
-
4
NONE
48
21
3
-
-
-
3
-
-
-
-
-
-
S
=
1
-
6
SHARED
55
28
1
-
1
-
-
-
-
-
-
-
-
B
=
2
-
2
BY
27
9
9
-
-
-
-
-
-
-
-
-
9
E
=
5
-
10
EVERYTHING
133
61
7
-
-
-
-
-
-
-
7
-
-
M
=
4
-
10
MULTIPLED
121
49
4
-
-
-
-
4
-
-
-
-
-
I
=
9
-
2
IN
23
14
5
-
-
-
-
-
5
-
-
-
-
A
=
1
-
9
ABUNDANCE
65
29
2
-
-
2
-
-
-
-
-
-
-
-
-
35
-
58
First Total
626
266
59
-
1
2
3
8
5
6
14
8
18
-
-
3+5
-
5+8
Add to Reduce
6+2+6
2+6+6
5+9
-
-
-
-
-
-
-
1+4
-
1+8
-
-
8
-
13
Second Total
14
14
10
-
1
2
3
8
5
6
5
8
9
-
-
-
-
1+3
Reduce to Deduce
1+4
1+4
1+0
-
-
-
-
-
-
-
-
-
-
-
-
8
-
4
Essence of Number
5
5
5
-
1
2
3
8
5
6
5
8
9

 

 

26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
9
-
-
-
-
5
6
-
-
-
1
-
-
-
-
6
-
8
+
=
43
4+3
=
7
=
7
=
7
-
-
-
-
-
-
-
-
8
9
-
-
-
-
14
15
-
-
-
19
-
-
-
-
24
-
26
+
=
115
1+1+5
=
7
=
7
=
7
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
-
-
1
2
3
4
-
-
7
8
9
-
2
3
4
5
-
7
-
+
=
83
8+3
=
11
1+1
2
=
2
-
1
2
3
4
5
6
7
-
-
10
11
12
13
-
-
16
17
18
-
20
21
22
23
-
25
-
+
=
236
2+3+6
=
11
1+1
2
=
2
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
+
=
351
3+5+1
=
9
=
9
=
9
-
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
+
=
126
1+2+6
=
9
=
9
=
9
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
+
=
1
occurs
x
3
=
3
=
3
-
-
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
+
=
2
occurs
x
3
=
6
=
6
-
-
-
3
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
3
-
-
-
-
-
+
=
3
occurs
x
3
=
9
=
9
-
-
-
-
4
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
4
-
-
-
-
+
=
4
occurs
x
3
=
12
1+2
3
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
-
+
=
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
-
-
+
=
6
occurs
x
3
=
18
1+8
9
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
7
-
+
=
7
occurs
x
3
=
21
2+1
3
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
+
=
8
occurs
x
3
=
24
2+4
6
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
+
=
9
occurs
x
2
=
18
1+8
9
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
45
-
-
26
-
126
-
54
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
4+5
-
-
2+6
-
1+2+6
-
5+4
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
9
-
-
8
-
9
-
9
-
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
-
-
-
-
-
-
-
-
-
-
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
9
-
-
8
-
9
-
9

 

 

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

 

I

SAY

GODS I ME I GODS

REAL REALITY REVEALED REALITY REAL

AND WOULD IT HAVE BEEN WORTH IT AFTER ALL, WHEN ALL IS SAID AND DONE, TO DENY A GOD INSPIRED MAGICIAN. THE SCATTERING OF BLESSED STAR STARRY STARDUST UPON AND AROUND THE MINDS I MINDS EYE. I ASK YOU, O DEAR GOOD AND TRUSTED FRIENDS, TO ACCEPT THESE EXALTED NAMES AS GODS HOLY ACCOLADES, ACCORDED OF GODS CREATORS. AND I HUMBLY REQUEST YOU TO PLEASE ALLOW ME, WITH KINDLY GOOD GRACE, AND AN UNDERSTANDING CREATIVE VISION. THE ODD LIGHT TOUCHES OF FAIRY STAR DUST. SUCH WORDS OF POWER AS ARE IMPORTANT TO THE GREAT WORK, AND DAVID IS OBLIGATED TO THE SYMBOLICAL MARKING OF THESE MAGICALLLY ESOTERIC, AND MYSTERIOUS WORDS OF POWER MANIFESTING FORTH FROM OUT THE IN OF ETERNAL MIND. MIRACULOUSLY BIRTHED OF THE GREAT COSMIC MOTHER. THE BE ALL AND END ALL OF EVERYTHING. THE HE AS IN SHE THAT IS ALWAYS FATHER TO THE THOUGHT. IN HOMAGE TO THAT. GODS EVER AND FOREVER LIVING CONSCIOUSNESS. THE PERFECT SUBLIME CREATIVE EXPRESSION OF THE DIVINE. THAT THAT THAT, HOLY, WHOLLY HOLY, ISISIS. BELOVED, WE ARE LIVING WITHIN THE MYTH OF MYTHS THE STORY OF THE PHYSICAL, OF MIND, AND SPIRIT. IN SACRED RECOGNITION OF KARMAS GLORIOUSLY MAGIC MIND. ACKNOWLEDGING ONCE AND FOR ALL SWEET ALCHERINGA, GODS HOLY DREAM TIME. THAT FECUND FERTILE MEMORY STORE FROM OUT THE IN OF WHICH IS KNIT THE SUM OF EVERLASTING LIFE. O DREAMER OF DREAMS, READ DEAR, READ THE MYTH OF N THE EVERLASTING NOW. ALL IS NOT WHAT IT SEEMS. DEPENDS ON YOUR I ME STATE OF MIND. THE SEE OF THE SELF THAT FEELS.

 

 

 

1
-
O
=
6
-
3
ONE
34
16
7
2
-
T
=
2
-
3
TWO
58
13
4
3
-
T
=
2
-
5
THREE
56
29
2
4
-
F
=
6
-
4
FOUR
60
24
6
5
-
F
=
6
-
4
FIVE
42
24
6
6
-
S
=
1
-
3
SIX
52
16
7
7
-
S
=
1
-
5
SEVEN
65
20
2
8
-
E
=
5
-
5
EIGHT
49
31
4
9
-
N
=
5
-
4
NINE
42
24
6
45
-
-
-
34
-
36
Add
458
197
44
4+5
-
-
-
3+4
-
3+6
Reduce
4+5+8
1+9+7
4+4
9
-
-
-
7
4
9
Deduce
17
17
8
-
-
-
-
-
-
-
Produce
1+7
1+7
-
9
-
-
-
7
-
9
Essence
8
8
8

 

 

0
-
Z
=
8
-
4
ZERO
64
28
1
1
-
O
=
6
-
3
ONE
34
16
7
2
-
T
=
2
-
3
TWO
58
13
4
3
-
T
=
2
-
5
THREE
56
29
2
4
-
F
=
6
-
4
FOUR
60
24
6
5
-
F
=
6
-
4
FIVE
42
24
6
6
-
S
=
1
-
3
SIX
52
16
7
7
-
S
=
1
-
5
SEVEN
65
20
2
8
-
E
=
5
-
5
EIGHT
49
31
4
9
-
N
=
5
-
4
NINE
42
24
6
45
-
-
-
42
-
40
Add
522
225
45
4+5
-
-
-
4+2
-
4+0
Reduce
5+2+2
2+2+5
4+5
9
-
-
-
6
-
4
Deduce
9
9
9

 

 

ADVENT 999 ADVENT

 

A

MAZE

IN

ZAZAZA ENTER AZAZAZ

AZAZAZAZAZAZAZZAZAZAZAZAZAZA

ZAZAZAZAZAZAZAZAZAAZAZAZAZAZAZAZAZAZ

THE

MAGICALALPHABET

ABCDEFGHIJKLMNOPQRSTUVWXYZZYXWVUTSRQPONMLKJIHGFEDCBA

 12345678910111213141516171819202122232425262625242322212019181716151413121110987654321

 

 

WORK DAYS OF GOD

Herbert W Morris D.D.circa 1883

Page 22

"As all the words in the English language are composed out of the twenty-six letters of the alphabet,.."

 

 

LIGHT AND LIFE

Lars Olof Bjorn 1976

Page 197

"By writing the 26 letters of the alphabet in a certain order one may put down almost any message (this book 'is written with the same letters' as the Encyclopaedia Britannica and Winnie the Pooh, only the order of the letters differs). In the same way Nature is able to convey with her language how a cell and a whole organism is to be constructed and how it is to function. Nature has succeeded better than we humans; for the genetic code there is only one universal language which is the same in a man, a bean plant and a bacterium."

"BY WRITING THE 26 LETTERS OF THE ALPHABET IN A CERTAIN ORDER

ONE MAY PUT DOWN ALMOST ANY MESSAGE"

 

 

A
B
C
D
E
F
G
H
I
1
2
3
4
5
6
7
8
9
 =
 =
 =
 =
 =
 =
 =
 =
=
 =
 =
 =
 =
 =
 =
 =
 =
=
J
K
L
M
N
O
P
Q
R
10
11
12
13
14
15
16
17
18
1+0
1+1
1+2
1+3
1+4
1+5
1+6
1+7
1+8
1
2
3
4
5
6
7
8
9
 =
 =
 =
 =
 =
 =
 =
 =
=
 =
 =
 =
 =
 =
 =
 =
 =
=
S
T
U
V
W
X
Y
Z
I
19
20
21
22
23
24
25
26
9
1+9
2+0
2+1
2+2
2+3
2+4
2+5
2+6
ME
1
2
3
4
5
6
7
8
9
 =
 =
 =
 =
 =
 =
 =
 =
=
 =
 =
 =
 =
 =
 =
 =
 =
=
I
ME
I
ME
I
ME
I
ME
I
9
18
9
18
9
18
9
18
9
=
1+8
=
1+8
=
1+8
=
1+8
=
=
9
=
9
=
9
=
9
=
I
ME
I
ME
I
ME
I
ME
1
9
9
9
9
9
9
9
9
9
I
ME
I
ME
I
ME
I
ME
1

 

 

"BY WRITING THE 26 LETTERS OF THE ALPHABET IN A CERTAIN ORDER

ONE MAY PUT DOWN ALMOST ANY MESSAGE"

 

 

A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
-
-
-
-
-
-
-
-
-
1+0
1+1
1+2
1+3
1+4
1+5
1+6
1+7
1+8
1+9
2+0
2+1
2+2
2+3
2+4
2+5
2+6
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
 -
-
-
-
-
-
-
-
-
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z

 

 

A

HISTORY OF GOD

Karen Armstrong 1993

The God of the Mystics

Page 250

"Perhaps the most famous of the early Jewish mystical texts is the fifth century Sefer Yezirah (The Book of Creation). There is no attempt to describe the creative process realistically; the account is unashamedly symbolic and shows God creating the world by means of language as though he were writing a book. But language has been entirely transformed and the message of creation is no longer clear. Each letter of the Hebrew alphabet is given a numerical value; by combining the letters with the sacred numbers, rearranging them in endless configurations, the mystic weaned his mind away from the normal connotations of words."

 

 

THERE IS NO ATTEMPT MADE TO DESCRIBE THE CREATIVE PROCESS REALISTICALLY

THE ACCOUNT IS SYMBOLIC AND SHOWS GOD CREATING THE WORLD BY MEANS OF LANGUAGE

AS THOUGH WRITING A BOOK BUT LANGUAGE ENTIRELY TRANSFORMED

THE MESSAGE OF CREATION IS CLEAR EACH LETTER OF

THE

ALPHABET

IS

GIVEN

A

NUMERICAL

VALUE BY COMBINING THE LETTERS WITH THE SACRED NUMBERS

REARRANGING THEM IN ENDLESS CONFIGURATIONS

THE MYSTIC WEANED THE MIND AWAY FROM THE NORMAL CONNOTATIONS OF WORDS

 

THIS IS THE SCENE OF THE SCENE UNSEEN

THE UNSEEN SEEN OF THE SCENE UNSEEN THIS IS THE SCENE

 

 

3
THE
33
15
6
4
MIND
40
22
4
2
OF
21
12
3
9
HUMANKIND
95
41
5
18
First Total
189
90
18
1+8
Add to Reduce
1+8+9
9+0
1+8
9
Second Total
18
9
9
-
Reduce to Deduce
1+8
-
-
9
Essence of Number
9
9
9

 

 

THE

FAR YONDER SCRIBE

AND OFT TIMES SHADOWED SUBSTANCES WATCHED IN FINE AMAZE

THE

ZED ALIZ ZED

IN

SWIFT REPEAT SCATTER STAR DUST AMONGST THE LETTERS OF THEIR PROGRESS

 

 

NUMBER

9

THE SEARCH FOR THE SIGMA CODE

Cecil Balmond 1998

Cycles and Patterns

Page 165

Patterns

"The essence of mathematics is to look for patterns.

Our minds seem to be organised to search for relationships and sequences. We look for hidden orders.

These intuitions seem to be more important than the facts themselves, for there is always the thrill at finding something, a pattern, it is a discovery - what was unknown is now revealed. Imagine looking up at the stars and finding the zodiac!

Searching out patterns is a pure delight.

Suddenly the counters fall into place and a connection is found, not necessarily a geometric one, but a relationship between numbers, pictures of the mind, that were not obvious before. There is that excitement of finding order in something that was otherwise hidden.

And there is the knowledge that a huge unseen world lurks behind the facades we see of the numbers themselves."

 

 

FINGERPRINTS OF THE GODS

A QUEST FOR THE BEGINNING AND THE END

Graham Hancock 1995

Chapter 32

Speaking to the Unborn

Page 285

"It is understandable that a huge range of myths from all over the ancient world should describe geological catastrophes in graphic detail. Mankind survived the horror of the last Ice Age, and the most plausible source for our enduring traditions of flooding and freezing, massive volcanism and devastating earthquakes is in the tumultuous upheavals unleashed during the great meltdown of 15,000 to 8000 BC. The final retreat of the ice sheets, and the consequent 300-400 foot rise in global sea levels, took place only a few thousand years before the beginning of the historical period. It is therefore not surprising that all our early civilizations should have retained vivid memories of the vast cataclysms that had terrified their forefathers.
Much harder to explain is the peculiar but distinctive way the myths of cataclysm seem to bear the intelligent imprint of a guiding hand.l Indeed the degree of convergence between such ancient stories is frequently remarkable enough to raise the suspicion that they must all have been 'written' by the same 'author'.
Could that author have had anything to do with the wondrous deity, or superhuman, spoken of in so many of the myths we have reviewed, who appears immediately after the world has been shattered by a horrifying geological catastrophe and brings comfort and the gifts of civilization to the shocked and demoralized survivors?
White and bearded, Osiris is the Egyptian manifestation of this / Page 286 / universal figure, and it may not be an accident that one of the first acts he is remembered for in myth is the abolition of cannibalism among the primitive inhabitants of the Nile Valley.2 Viracocha, in South America, was said to have begun his civilizing mission immediately after a great flood; Quetzalcoatl, the discoverer of maize, brought the benefits of crops, mathematics, astronomy and a refined culture to Mexico after the Fourth Sun had been overwhelmed by a destroying deluge.
Could these strange myths contain a record of encounters between scattered palaeolithic tribes which survived the last Ice Age and an as yet unidentified high civilization which passed through the same epoch?
And could the myths be attempts to communicate?

A message in the bottle of time"

'Of all the other stupendous inventions,' Galileo once remarked,

what sublimity of mind must have been his who conceived how to communicate his most secret thoughts to any other person, though very distant either in time or place, speaking with those who are in the Indies, speaking to those who are not yet born, nor shall be this thousand or ten thousand years? And with no greater difficulty than the various arrangements of two dozen little signs on paper? Let this be the seal of all the admirable inventions of men.3

If the 'precessional message' identified by scholars like Santillana, von Dechend and Jane Sellers is indeed a deliberate attempt at communication by some lost civilization of antiquity, how come it wasn't just written down and left for us to find? Wouldn't that have been easier than encoding it in myths? Perhaps.
Nevertheless, suppose that whatever the message was written on got destroyed or worn away after many thousands of years? Or suppose that the language in which it was inscribed was later forgotten utterly (like the enigmatic Indus Valley script, which has been studied closely for more than half a century but has so far resisted all attempts at decoding)? It must be obvious that in such circumstances a written / Page 287 / legacy to the future would be of no value at all, because nobody would be able to make sense of it.
What one would look for, therefore, would be a universal language, the kind of language that would be comprehensible to any technologically advanced society in any epoch, even a thousand or ten thousand years into the future. Such languages are few and far between, but mathematics is one of them - and the city of Teotihuacan may be the calling-card of a lost civilization written in the eternal language of mathematics.
Geodetic data, related to the exact positioning of fixed geographical points and to the shape and size of the earth, would also remain valid and recognizable for tens of thousands of years, and might be most conveniently expressed by means of cartography (or in the construction of giant geodetic monuments like the Great Pyramid of Egypt, as we shall see).
Another 'constant' in our solar system is the language of time: the great but regular intervals of time calibrated by the inch-worm creep of precessional motion. Now, or ten thousand years in the future, a message that prints out numbers like 72 or 2160 or 4320 or 25,920 should be instantly intelligible to any civilization that has evolved a modest talent for mathematics and the ability to detect and measure the almost imperceptible reverse wobble that the sun appears to make along the ecliptic against the background of the fixed stars..."

"What one would look for, therefore, would be a universal language, the kind of language that would be comprehensible to any technologically advanced society in any epoch, even a thousand or ten thousand years into the future. Such languages are few and far between, but mathematics is one of them"

"WRITTEN IN THE ETERNAL LANGUAGE OF MATHEMATICS"

 

 

 

THE SCULPTURE OF VIBRATIONS 1971

 

 

-
THE RAINBOW LIGHT
-
-
-
3
THE
33
15
6
7
RAINBOW
82
37
1
5
LIGHT
56
29
2
15
THE RAINBOW LIGHT
171
81
9
1+5
-
1+7+1
8+1
-
6
THE RAINBOW LIGHT
9
9
9

 

 

15
THE RAINBOW LIGHT
-
-
-
-
THE
33
15
6
-
R
18
9
9
-
A
1
1
1
-
I
9
9
9
-
N+B+O+W
54
18
9
-
L
12
3
3
-
I
9
9
9
-
G+H+T
35
17
8
15
THE RAINBOW LIGHT
171
81
54
1+5
-
1+7+1
8+1
5+4
6
THE RAINBOW LIGHT
9
9
9

 

 

THE LIGHT IS RISING RISING IS THE LIGHT

 

 

26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
+
=
351
3+5+1
=
9
-
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
+
=
126
1+2+6
=
9
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-

 

 

9
A
B
C
D
E
F
G
H
I
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
9
+
=
17
1+7
=
8
=
8
-
-
-
-
-
-
-
-
8
9
+
=
17
1+7
=
8
=
8
9
A
B
C
D
E
F
G
H
I
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
-
-
+
=
28
2+8
=
10
1+0
1
-
1
2
3
4
5
6
7
-
-
+
=
28
2+8
=
10
1+0
1
9
A
B
C
D
E
F
G
H
I
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
+
=
45
4+5
=
9
=
9
-
1
2
3
4
5
6
7
8
9
+
=
45
4+5
=
9
=
9
9
A
B
C
D
E
F
G
H
I
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
+
=
1
occurs
x
1
=
1
-
-
2
-
-
-
-
-
-
-
+
=
2
occurs
x
1
=
2
-
-
-
3
-
-
-
-
-
-
+
=
3
occurs
x
1
=
3
-
-
-
-
4
-
-
-
-
-
+
=
4
occurs
x
1
=
4
-
-
-
-
-
5
-
-
-
-
+
=
5
occurs
x
1
=
5
-
-
-
-
-
-
6
-
-
-
+
=
6
occurs
x
1
=
6
-
-
-
-
-
-
-
7
-
-
+
=
7
occurs
x
1
=
7
-
-
-
-
-
-
-
-
8
-
+
=
8
occurs
x
1
=
8
-
-
-
-
-
-
-
-
-
9
+
=
9
occurs
x
1
=
9
9
A
B
C
D
E
F
G
H
I
-
-
45
-
-
9
-
45
-
1
2
3
4
5
6
7
8
9
-
-
4+5
-
-
-
-
4+5
9
A
B
C
D
E
F
G
H
I
-
-
9
-
-
9
-
9
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
9
A
B
C
D
E
F
G
H
I
-
-
9
-
-
9
-
9

 

 

9
J
K
L
M
N
O
P
Q
R
-
-
-
-
-
-
-
-
-
-
-
-
-
5
6
-
-
-
+
=
11
1+1
=
2
=
2
-
-
-
-
-
14
15
-
-
-
+
=
29
2+9
=
11
1+1
2
9
J
K
L
M
N
O
P
Q
R
-
-
-
-
-
-
-
-
-
1
2
3
4
-
-
7
8
9
+
=
34
3+4
=
7
=
7
-
10
11
12
13
-
-
16
17
18
+
=
97
9+7
=
16
1+6
7
9
J
K
L
M
N
O
P
Q
R
-
-
-
-
-
-
-
-
-
10
11
12
13
14
15
16
17
18
+
=
126
1+2+6
=
9
=
9
-
1
2
3
4
5
6
7
8
9
+
=
45
4+5
=
9
=
9
9
J
K
L
M
N
O
P
Q
R
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
+
=
1
occurs
x
1
=
1
-
-
2
-
-
-
-
-
-
-
+
=
2
occurs
x
1
=
2
-
-
-
3
-
-
-
-
-
-
+
=
3
occurs
x
1
=
3
-
-
-
-
4
-
-
-
-
-
+
=
4
occurs
x
1
=
4
-
-
-
-
-
5
-
-
-
-
+
=
5
occurs
x
1
=
5
-
-
-
-
-
-
6
-
-
-
+
=
6
occurs
x
1
=
6
-
-
-
-
-
-
-
7
-
-
+
=
7
occurs
x
1
=
7
-
-
-
-
-
-
-
-
8
-
+
=
8
occurs
x
1
=
8
-
-
-
-
-
-
-
-
-
9
+
=
9
occurs
x
1
=
9
9
J
K
L
M
N
O
P
Q
R
-
-
45
-
-
9
-
45
-
10
11
12
13
14
15
16
17
18
-
-
4+5
-
-
-
-
4+5
9
J
K
L
M
N
O
P
Q
R
-
-
9
-
-
9
-
9
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
9
J
K
L
M
N
O
P
Q
R
-
-
9
-
-
9
-
9

 

 

8
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
1
-
-
-
-
6
-
8
+
=
15
1+5
=
6
=
6
-
19
-
-
-
-
24
-
26
+
=
69
6+9
=
15
1+5
6
8
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
2
3
4
5
-
7
-
+
=
21
2+1
=
3
=
3
-
-
20
21
22
23
-
25
-
+
=
111
1+1+1
=
3
=
3
8
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
19
20
21
22
23
24
25
26
+
=
 180
1+8+0
=
9
=
9
-
1
2
3
4
5
6
7
8
+
=
36
3+6
=
9
=
9
8
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
+
=
1
occurs
x
1
=
1
-
-
2
-
-
-
-
-
-
+
=
2
occurs
x
1
=
2
-
-
-
3
-
-
-
-
-
+
=
3
occurs
x
1
=
3
-
-
-
-
4
-
-
-
-
+
=
4
occurs
x
1
=
4
-
-
-
-
-
5
-
-
-
+
=
5
occurs
x
1
=
5
-
-
-
-
-
-
6
-
-
+
=
6
occurs
x
1
=
6
-
-
-
-
-
-
-
7
-
+
=
7
occurs
x
1
=
7
-
-
-
-
-
-
-
-
8
+
=
8
occurs
x
1
=
8
8
S
T
U
V
W
X
Y
Z
-
-
36
-
-
8
-
36
-
1
2
3
4
5
6
7
8
-
-
3+6
-
-
-
-
3+6
8
S
T
U
V
W
X
Y
Z
-
-
9
-
-
8
-
9
-
19
20
21
22
23
24
25
26
-
-
-
-
-
-
-
-
8
S
T
U
V
W
X
Y
Z
-
-
9
-
-
8
-
9

 

 

26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
9
-
-
-
-
5
6
-
-
-
1
-
-
-
-
6
-
8
+
=
43
4+3
=
7
=
7
=
7
-
-
-
-
-
-
-
-
8
9
-
-
-
-
14
15
-
-
-
19
-
-
-
-
24
-
26
+
=
115
1+1+5
=
7
=
7
=
7
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
-
-
1
2
3
4
-
-
7
8
9
-
2
3
4
5
-
7
-
+
=
83
8+3
=
11
1+1
2
=
2
-
1
2
3
4
5
6
7
-
-
10
11
12
13
-
-
16
17
18
-
20
21
22
23
-
25
-
+
=
236
2+3+6
=
11
1+1
2
=
2
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
+
=
351
3+5+1
=
9
=
9
=
9
-
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
+
=
126
1+2+6
=
9
=
9
=
9
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
+
=
1
occurs
x
3
=
3
=
3
-
-
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
+
=
2
occurs
x
3
=
6
=
6
-
-
-
3
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
3
-
-
-
-
-
+
=
3
occurs
x
3
=
9
=
9
-
-
-
-
4
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
4
-
-
-
-
+
=
4
occurs
x
3
=
12
1+2
3
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
-
+
=
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
-
-
+
=
6
occurs
x
3
=
18
1+8
9
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
7
-
+
=
7
occurs
x
3
=
21
2+1
3
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
+
=
8
occurs
x
3
=
24
2+4
6
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
+
=
9
occurs
x
2
=
18
1+8
9
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
45
-
-
26
-
126
-
54
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
4+5
-
-
2+6
-
1+2+6
-
5+4
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
9
-
-
8
-
9
-
9
-
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
-
-
-
-
-
-
-
-
-
-
26
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
-
-
9
-
-
8
-
9
-
9

 

 

 

 

 

 

THE DEATH OF GODS IN ANCIENT EGYPT

Jane B. Sellars 1992

Page 204

"The overwhelming awe that accompanies the realization, of the measurable orderliness of the universe strikes modern man as well. Admiral Weiland E. Byrd, alone In the Antarctic for five months of polar darkness, wrote these phrases of intense feeling:

Here were the imponderable processes and forces of the cosmos, harmonious and soundless. Harmony, that was it! I could feel no doubt of oneness with the universe. The conviction came that the rhythm was too orderly. too harmonious, too perfect to be a product of blind chance - that, therefore there must be purpose in the whole and that man was part of that whole and not an accidental offshoot. It was a feeling that transcended reason; that went to the heart of man's despair and found it groundless. The universe was a cosmos, not a chaos; man was as rightfully a part of that cosmos as were the day and night.10

Returning to the account of the story of Osiris, son of Cronos god of' Measurable Time, Plutarch takes, pains to remind the reader of the original Egyptian year consisting of 360 days.

Phrases are used that prompt simple mental. calculations and an attention to numbers, for example, the 360-day year is described as being '12 months of 30 days each'. Then we are told that, Osiris leaves on a long journey, during which Seth, his evil brother, plots with 72 companions to slay Osiris: He also secretly obtained the measure of Osiris and made ready a chest in which to entrap him.

The, interesting thing about this part of the-account is that nowhere in the original texts of the Egyptians are we told that Seth, has 72 companions. We have already been encouraged to equate Osiris with the concept of measured time; his father being Cronos. It is also an observable fact that Cronos-Saturn has the longest sidereal period of the known planets at that time, an orbit. of 30 years. Saturn is absent from a specific constellation for that length of time.

A simple mathematical fact has been revealed to any that are even remotely sensitive to numbers: if you multiply 72 by 30, the years of Saturn's absence (and the mention of Osiris's absence prompts one to recall this other), the resulting product is 2,160: the number of years required, for one 30° shift, or a shift: through one complete sign of the zodiac. This number multplied by the /Page205 / 12 signs also gives 25,920. (And Plutarch has reminded us of 12)

If you multiply the unusual number 72 by 360, a number that Plutarch mentions several times, the product will be 25,920, again the number of years symbolizing the ultimate rebirth.

This 'Eternal Return' is the return of, say, Taurus to the position of marking the vernal equinox by 'riding in the solar bark with. Re' after having relinquished this honoured position to Aries, and subsequently to the to other zodiacal constellations.

Such a return after 25,920 years is indeed a revisit to a Golden Age, golden not only because of a remarkable symmetry In the heavens, but golden because it existed before the Egyptians experienced heaven's changeability.

But now to inform the reader of a fact he or she may already know. Hipparaus did: not really have the exact figures: he was a trifle off in his observations and calculations. In his published work, On the Displacement of the Solstitial and Equinoctial Signs, he gave figures of 45" to 46" a year, while the truer precessional lag along the ecliptic is about 50 seconds. The exact measurement for the lag, based on the correct annual lag of 50'274" is 1° in 71.6 years, or 360° in 25,776 years, only 144 years less than the figure of 25,920.

With Hipparchus's incorrect figures a 'Great Year' takes from 28,173.9 to 28,800 years, Incorrect by a difference of from 2,397.9 years to 3,024.

Since Nicholas Copernicus (AD 1473-1543) has always been credited with giving the correct numbers (although Arabic astronomer Nasir al-Din Tusi,11 born AD 1201, is known to have fixed the Precession at 50°), we may correctly ask, and with justifiable astonishment 'Just whose information was Plutarch transmitting'

AN IMPORTANT POSTSCRIPT

Of course, using our own notational system, all the important numbers have digits that reduce to that amazing number 9 a number that has always delighted budding mathematician.

Page 206

Somewhere along the way, according to Robert Graves, 9 became the number of lunar wisdom.12

This number is found often in the mythologies of the world. the Viking god Odin hung for nine days and nights on the World Tree in order to acquire the secret of the runes, those magic symbols out of which writing and numbers grew. Only a terrible sacrifice would give away this secret, which conveyed upon its owner power and dominion over all, so Odin hung from his neck those long 9 days and nights over the 'bottomless abyss'. In the tree were 9 worlds, and another god was said to have been born of 9 mothers.

Robert Graves, in his White Goddess, Is intrigued by the seemingly recurring quality of the number 72 in early myth and ritual. Graves tells his reader that 72 is always connected with the number 5, which reflects, among other things, the five Celtic dialects that he was investigating. Of course, 5 x 72= 360, 360 x 72= 25,920. Five is also the number of the planets known to the ancient world, that is, Saturn, Jupiter, Mars, Venus Mercury.

Graves suggests a religious mystery bound up with two ancient Celtic 'Tree Alphabets' or cipher alphabets, which as genuine articles of Druidism were orally preserved and transmitted for centuries. He argues convincingly that the ancient poetry of Europe was ultimately based on what its composers believed to be magical principles, the rudiments of which formed a close religious secret for centuries. In time these were-garbled, discredited and forgotten.

Among the many signs of the transmission of special numbers he points out that the aggregate number of letter strokes for the complete 22-letter Ogham alphabet that he is studying is 72 and that this number is the multiple of 9, 'the number of lunar wisdom'. . . . he then mentions something about 'the seventy day season during which Venus moves successively from. maximum eastern elongation 'to inferior conjunction and maximum western elongation'.13

Page 207

"...Feniusa Farsa, Graves equates this hero with Dionysus Farsa has 72 assistants who helped him master the 72 languages created at the confusion of Babel, the tower of which is said to be built of 9 different materials

We are also reminded of the miraculous translation into Greek of the Five Books of Moses that was done by 72 scholars working for 72 days, Although the symbol for the Septuagint is LXX, legend, according to the fictional letter of Aristeas, records 72. The translation was done for Ptolemy Philadelphus (c.250 BC), by Hellenistic Jews, possibly from Alexandra.14

Graves did not know why this number was necessary, but he points out that he understands Frazer's Golden Bough to be a a book hinting that 'the secret involves the truth that the Christian dogma, and rituals, are the refinement of a great body of primitive beliefs, and that the only original element in Christianity- is the personality of Christ.15

Frances A. Yates, historian of Renaissance hermetisma tells, us the cabala had 72 angels through which the sephiroth (the powers of God) are believed to be approached, and further, she supplies the information that although the Cabala supplied a set of 48 conclusions purporting to confirm the Christian religion from the foundation of ancient wisdom, Pico Della Mirandola, a Renaissance magus, introduced instead 72, which were his 'own opinion' of the correct number. Yates writes, 'It is no accident there are seventy-two of Pico's Cabalist conclusions, for the conclusion shows that he knew something of the mystery of the Name of God with seventy-two letters.'16

In Hamlet's Mill de Santillarta adds the facts that 432,000 is the number of syllables in the Rig-Veda, which when multiplied by the soss (60) gives 25,920" (The reader is forgiven for a bit of laughter at this point)

Thee Bible has not escaped his pursuit. A prominent Assyriologist of the last century insisted that the total of the years recounted
mounted in Genesis for the lifetimes of patriarchs from the Flood also contained the needed secret numbers. (He showed that in the 1,656.years recounted in the Bible there are 86,400 7 day weeks, and dividing this number yields / Page 208 / 43,200.) In Indian yogic schools it is held that all living beings exhale and inhale 21,600 times a day, .multiply this by 2 and again we have.the necessary 432 digits.

Joseph Campbell discerns the secret in the date set for the coming of Patrick to Ireland. Myth-gives this date-as.- the interest-
ing number of AD.432.18

Whatever one may think-of some of these number coincidences, it becomes. difficult to escape the suspicion that many signs (number and otherwise) -indicate that early man observed the results.. of the movement of Precession . and that the-.transmission of this information was .considered of prime importance.

'With the awareness of the phenomenon, observers would certainly have tried for its measure, and such an endeavour would
have constituted the construction-of a 'Unified Field Theory' for nothing .less than Creation itself. Once determined, it would have been information worthy of secrecy and worthy of the passing on to future adepts.

But one last word about mankind's romance with number coincidences.The antagonist in John Updike's novel, Roger's Version, is a computer hacker, who, convinced.,that scientific evidence of God's existence is accumulating, endeavours to prove it by feeding -all the available scientific information. into a comuter. In his search for God 'breaking, through', he has become fascinated by certain numbers that have continually been cropping up. He explains them excitedly as 'the terms of Creation':

"...after a while I noticed that all over the sheet there seemed to hit these twenty-fours Jumping out at me. Two four; two,four.Planck time, for instance, divided by the radiation constant yields a figure near eight times ten again to the negative twenty-fourth, and the permittivity of free space, or electric constant, into the Bohr radiusekla almost exactly six times ten to the negative twenty-fourth. On positive side, the electromagnetic line-structure constant times Hubble radius - that is, the size of the universe as we now perceive it gives us something quite close to ten to the twenty-fourth, and the strong-force constant times the charge on the proton produces two point four times ten to the negative eighteenth, for another I began to circle twenty-four wherever it appeared on the Printout here' - he held it up. his piece of striped and striped wallpaper, decorated / Page 209 / with a number of scarlet circles - 'you can see it's more than random.'19
This inhabitant of the twentieth century is convinced that the striking occurrences of 2 and 4 reveal the sacred numbers by which God is speaking to us.

So much for any scorn directed to ancient man's fascination with number coincidences. That fascination is alive and well, Just a bit more incomprehensible"

 

 

WE ARE THE DEAD SHORT TIME AGO WE LIVED FELT DAWN SAW SUNSET GLOW AND NOW

?

 

 

-
6
A
T
U
M
-
R
A
-
-
-
-
-
--
-
-
-
`-
1
20
21
13
-
18
1
+
=
74
7+4
=
11
1+1
2
-
-
1
2
3
4
-
9
1
+
=
20
2+0
=
2
-
2
-
6
A
T
U
M
-
R
A
-T
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
-
-
1
occurs
x
2
=
2
-
-
-
2
-
-
-
-
-
-
-
2
occurs
x
1
=
2
-
-
-
-
3
-
-
-
-
-
-
3
occurs
x
1
=
3
-
-
-
-
-
4
-
-
-
-
-
4
occurs
x
1
=
4
5
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
26
6
A
T
U
M
-
R
A
-
-
19
-
-
6
-
20
2+6
-
-
-
-
-
-
9
-
-
-
1+9
-
-
-
-
2+0
8
6
A
T
U
M
-
R
A
-
-
10
-
-
6
-
2
-
-
-
-
-
-
-
-
-
-
-
1+0
-
-
-
-
-
8
6
A
T
U
M
-
R
A
-
-
1
-
-
6
-
2

 

 

6
A
T
U
M
-
R
A
-
-
-
-
-
--
-
-
`-
1
20
21
13
-
18
1
+
=
74
7+4
=
11
1+1
2
-
1
2
3
4
-
9
1
+
=
20
2+0
=
2
-
2
6
A
T
U
M
-
R
A
-T
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
-
-
1
occurs
x
2
=
2
-
-
2
-
-
-
-
-
-
-
2
occurs
x
1
=
2
-
-
-
3
-
-
-
-
-
-
3
occurs
x
1
=
3
-
-
-
-
4
-
-
-
-
-
4
occurs
x
1
=
4
-
-
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
6
A
T
U
M
-
R
A
-
-
19
-
-
6
-
20
-
-
-
-
-
-
9
-
-
-
1+9
-
-
-
-
2+0
6
A
T
U
M
-
R
A
-
-
10
-
-
6
-
2
-
1
2
3
4
-
9
1
-
-
1+0
-
-
-
-
-
6
A
T
U
M
-
R
A
-
-
1
-
-
6
-
2

 

 

6
A
T
U
M
R
A
-
-
-
-
-
--
-
-
`-
1
20
21
13
18
1
+
=
74
7+4
=
11
1+1
2
-
1
2
3
4
9
1
+
=
20
2+0
=
2
-
2
6
A
T
U
M
R
A
-T
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
1
occurs
x
2
=
2
-
-
2
-
-
-
-
-
-
2
occurs
x
1
=
2
-
-
-
3
-
-
-
-
-
3
occurs
x
1
=
3
-
-
-
-
4
-
-
-
-
4
occurs
x
1
=
4
-
-
-
-
-
9
-
-
-
9
occurs
x
1
=
9
6
A
T
U
M
R
A
-
-
19
-
-
6
-
20
-
-
-
-
-
9
-
-
-
1+9
-
-
-
Q
2+0
6
A
T
U
M
R
A
-
-
10
-
-
6
-
2
-
1
2
3
4
9
1
-T
-
1+0
-
-
-
-
-
6
A
T
U
M
R
A
-
-
1
-
-
6
-
2

 

 

-
4
A
T
U
M
-
-
-
-
-
--
-
-
-
`-
1
20
21
13
+
=
55
5+5
=
10
1+0
1
-
-
1
2
3
4
+
=
10
1+0
=
1
-
1
-
4
A
T
U
M
-T
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
1
=
1
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
-
-
-
-
3
-
-
-
3
occurs
x
1
=
3
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
5
-
-
-
-
-
-
-
5
-
-
-
-
-
6
-
-
-
-
-
-
-
6
-
-
-
-
-
7
-
-
-
-
-
-
-
7
-
-
-
-
-
8
-
-
-
-
-
-
-
8
-
-
-
-
-
9
-
-
-
-
-
-
-
9
-
-
-
-
-
35
4
A
T
U
M
-
-
10
-
-
4
-
10
3+5
-
-
-
-
-
-
-
1+0
-
-
-
Q
1+0
8
4
A
T
U
M
-
-
1
-
-
4
-
2

 

 

4
A
T
U
M
-
-
-
-
-
--
-
-
`-
1
20
21
13
+
=
55
5+5
=
10
1+0
1
-
1
2
3
4
+
=
10
1+0
=
1
-
1
4
A
T
U
M
-T
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
1
=
1
-
-
2
-
-
-
-
2
occurs
x
1
=
2
-
-
-
3
-
-
-
3
occurs
x
1
=
3
-
-
-
-
4
-
-
4
occurs
x
1
=
4
4
A
T
U
M
-
-
10
-
-
4
-
10
-
-
-
-
-
-
-
1+0
-
-
-
Q
1+0
4
A
T
U
M
-
-
1
-
-
4
-
2

 

 

OF TIME AND STARS

Arthur C. Clarke 1972

FOREWORD

"'Into the Comet' and 'The Nine Billion Names of God' both involve computers and the troubles they may cause us. While writing this preface, I had occasion to call upon my own HP 9100A computer, Hal Junior, to answer an interesting question. Looking at my records, I find that I have now written just about one hundred short stories. This volume contains eighteen of them: therefore, how many possible 18-story collections will I be able to put together? The answer ­as I am sure will be instantly obvious to you - is 100 x 99. . . x 84 x 83 divided by 18 x 17 x 16 ... x .2 x 1. This is an impressive number - Hal Junior tells me that it is approximately 20,772,733,124,605,000,000.

 

3
THE
33
15
6
4
NINE
42
24
6
7
BILLION
73
37
1
5
NAMES
52
16
7
2
OF
21
12
3
3
GOD
26
17
8
24
-
247
121
31
2+4
-
2+4+7
1+2+1
3+1
6
-
13
4
4
-
-
1+3
-
-
6
-
4
4
4

Page 15

The Nine Billion Names of God

 


'This is a slightly unusual request,' said Dr Wagner, with what he hoped was commendable restraint. 'As far as I know, it's the first time anyone's been asked to supply a Tibetan monastery with an Automatic Sequence Computer. I don't wish to be inquisitive, but I should hardly have thought that your - ah - establishment had much use for such a machine. Could you explain just what you intend to do with it?'
'Gladly,' replied the lama, readjusting his silk robes and carefully putting away the slide rule he had been using far currency conversions. 'Your Mark V Computer can carry out any routine mathematical operation involving up to ten digits. However, for our work we are interested in letters, not numbers. As we wish you to modify the output circuits, the machine will be printing words, not columns of figures.'
'I don't quite understand. . .'
'This is a project on which we have been working for the last three centuries - since the lamasery was founded, in fact. It is somewhat alien to your way of thought, so I hope you will listen with an open mind while I explain it.'
'Naturally.'
'It is really quite simple. We have been compiling a list which shall contain all the possible names of God.'
'I beg your pardon?'

Page16

'We have reason to believe,' continued the lama imperturbably, 'that all such names can be written with not more than nine letters in an alphabet we have devised.'
'And you have been doing this for three centuries?'
'Yes: we expected it would take us about fifteen thousand years to complete the task.'
'Oh,' Dr Wagner looked a little dazed. 'Now I see why you wanted to hire one of our machines. But what exactly is the purpose of this project?'
The lama hesitated for a fraction of a second, and Wagner wondered if he had offended him. If so, there was no trace of annoyance in the reply.
'Call it ritual, if you like, but it's a fundamental part of our belief. All the many names of the Supreme Being - God Jehova, Allah, and so on - they are only man-made labels. There is a philosophical problem of some difficulty here, which I do not propose to discuss, but somewhere among all the possible combinations of letters that can occur are what one may call the real names of God. By systematic permutation of letters, we have been trying to list them all.'
'I see. You've been starting at AAAAAAA . . . and working up to ZZZZZZZZ . . .'
'Exactly - though we use a special alphabet of our own. Modifying the electromatic typewriters to deal with this is, of course, trivial. A rather more interesting problem is that of devising suitable circuits to eliminate ridiculous combinations. For example, no letter must occur more than three times in succession.'
,'Three? Surely you mean two.'
'Three is correct: I am afraid it would take too long to explain why, even if you understood our language.' "

Page 68

Into the Comet


"Pickett's fingers danced over the beads, sliding them up and down the wires with lightning speed. There were twelve wires in all, so that the abacus could handle numbers up to 999,999,999,999 - or could be divided into separate sections where several independent calculations could be carried out simultaneously.
'374072,' said Pickett, after an incredibly brief interval of time. 'Now see how long you take to do it, with pencil and paper.'
There was a much longer delay before Martens, who like most mathematicians was poor at arithmetic, called out '375072'. A hasty check soon confirmed that Martens had taken at least three times as long as Pickett to arrive at the wrong answer.
The atronomer's face was a study in mingled chagrin, astonishment, and curiosity.
'Where did you learn that trick?' he asked. 'I thought those things could only add and subtract.'
'Well - multiplication's only repeated addition, isn't it? All I did was to add 856 seven times in the unit column, three times in the tens column, and four times in the hundreds column. You do the same thing when you use pencil and paper. Of course, there are some short cuts, but if you think I'm fast, you should have seen my granduncle. He used to work in a Yokohama bank, and you couldn't see his fingers / Page 69 / when he was going at speed"

 

 

I

SAY

THREAD THAT THREAD

THREAD READ DEATH DEATH READ THREAD

THREAD R DEATH DEATH R THREAD

THREAD READ DEAR THREAD

 

 

THE NEW ELIZABETHAN

REFERENCE DICTIONARY

An up-to-date vocabulary of the living English language

Circa 1900

FOURTH EDITION

Page 1472

thread (thred) [A.-S. thraed, from thrawan, to THROW (cp. Dut. draad, G. draht, Icel. thrathr)], n. A slender cord consisting of two or more yarns doubled or twisted ; a single filament of cotton, silk, wool, etc., esp. Lisle thread ; anything resembling this ; a fine line of colour etc. ; a thin seam or vein ; the spiral on a screw ; (fig.) a continuous course (of life etc.). v.t. To pass a thread through the eye or aperture of ; to string (beads etc.) on a thread ; (fig.) to pick (one's way) or to go through an intricate or crowded place, etc. ; to streak (the hair) with grey etc. ; to cut a thread on (a screw). thread and thrum : Good and bad together, all alike. threadbare, a. Worn so that the thread is visible, having the nap worn off ; (fig.) worn, trite, hackneyed. threadbareness, n. thread-mark, n. A mark produced by coloured silk fibres in banknotes to prevent counterfeiting. thread-paper, n. Soft paper for wrapping up thread, thread-worm, n. A thread-like nematode worm, esp. one infesting the rectum of children. threader, n. threadlike, a. and adv. thready, a. threadiness, n.

 

 

THE NEW ELIZABETHAN

REFERENCE DICTIONARY

An up-to-date vocabulary of the living English language

FOURTH EDITION

Circa 1900

Page 1472

thread (thred) [A.-S. thraed, from thrawan, to THROW (cp. Dut. draad, G. draht, Icel. thrathr)], n. A slender cord consisting of two or more yarns doubled or twisted ; a single filament of cotton, silk, wool, etc., esp. Lisle thread ; anything resembling this ; a fine line of colour etc. ; a thin seam or vein ; the spiral on a screw ; (fig.) a continuous course (of life etc.). v.t. To pass a thread through the eye or aperture of ; to string (beads etc.) on a thread ; (fig.) to pick (one's way) or to go through an intricate or crowded place, etc. ; to streak (the hair) with grey etc. ; to cut a thread on (a screw). thread and thrum : Good and bad together, all alike. threadbare, a. Worn so that the thread is visible, having the nap worn off ; (fig.) worn, trite, hackneyed. threadbareness, n. thread-mark, n. A mark produced by coloured silk fibres in banknotes to prevent counterfeiting. thread-paper, n. Soft paper for wrapping up thread, thread-worm, n. A thread-like nematode worm, esp. one infesting the rectum of children. threader, n. threadlike, a. and adv. thready, a. threadiness, n.

 

 

lisle thread: lisle thread

A strong tightly twisted cotton thread (usually made of long-staple cotton) - lisle. Derived forms: lisle threads. Type of: cotton. Nearest ... www.wordwebonline.com/en/LISLETHREAD

 

 

Definition - of Lisle from Dictionary.net

Lisle thread, a hard twisted cotton thread, originally produced at Lisle. Source: Webster's Revised Unabridged Dictionary (1913) ... www.dictionary.net/lisle - 9k

 

 

CASSELL'S ENGLISH DICTIONARY

1974

Lisle thread (lil thred) [ town in France, now Lille], n, A fine, hard thread orig. made at Lille.

 

 

-
16
H
A
N
G
I
N
G
-
B
Y
-
A
-
T
H
R
E
A
D
-
-
-
-
-
--
-
-
-
-
-
-
8
-
5
-
9
5
-
-
-
-
-
-
-
-
8
-
-
-
-
+
=
35
3+5
=
8
-
8
-
8
-
-
8
-
14
-
9
14
-
-
-
-
-
-
-
-
8
-
-
-
-
+
=
53
5+3
=
8
-
8
-
8
-
16
H
A
N
G
I
N
G
-
B
Y
-
A
-
T
H
R
E
A
D
-
-
-
-
-
--
-
-
-
-
-
-
-
1
-
7
-
-
7
-
2
7
-
1
-
2
-
9
5
1
4
+
=
46
4+6
=
10
1+0
1
-
1
-
-
-
1
-
7
-
-
7
-
2
25
-
1
-
20
-
18
5
1
4
+
=
91
9+1
=
10
1+0
1
-
1
-
16
H
A
N
G
I
N
G
-
B
Y
-
A
-
T
H
R
E
A
D
-
-
-
-
-
--
-
-
-
-
-
-
8
1
14
7
9
14
7
-
2
25
-
1
-
20
8
18
5
1
4
+
=
144
1+4+4
=
9
-
9
-
9
-
-
8
1
5
7
9
5
7
-
2
7
-
1
-
2
8
9
5
1
4
+
=
81
8+1
=
9
1+0
9
-
9
-
16
H
A
N
G
I
N
G
-
B
Y
-
A
-
T
H
R
E
A
D
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
-
-
-
1
occurs
x
3
=
3
-
3
``-
``-
-
-
-
-
-
-
-
-
2
-
-
-
-
2
-
-
-
-
-
-
-
2
occurs
x
2
=
4
-
4
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
4
-
-
-
-
5
-
-
5
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
5
occurs
x
3
=
15
1+5
6
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
7
-
-
7
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
3
=
21
2+1
3
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
8
occurs
x
2
=
16
1+6
7
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
9
16
H
A
N
G
I
N
G
-
B
Y
-
A
-
T
H
R
E
A
D
-
-
36
-
-
16
-
81
-
36
-
1+6
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
3+6
-
-
1+6
-
8+1
-
3+6
9
7
H
A
N
G
I
N
G
-
B
Y
-
A
-
T
H
R
E
A
D
-
-
9
-
-
7
-
9
-
9
-
-
8
1
5
7
9
5
7
-
2
7
-
1
-
2
8
9
5
1
4
-
-
-
-
-
-
-
-
-
-
9
7
H
A
N
G
I
N
G
-
B
Y
-
A
-
T
H
R
E
A
D
-
-
9
-
-
7
-
9
-
9

 

 

6
H
A
N
G
I
N
G
-
B
Y
-
A
-
T
H
R
E
A
D
-
-
-
-
-
--
-
-
-
-
-
8
-
5
-
9
5
-
-
-
-
-
-
-
-
8
-
-
-
-
+
=
35
3+5
=
8
-
8
-
8
-
8
-
14
-
9
14
-
-
-
-
-
-
-
-
8
-
-
-
-
+
=
53
5+3
=
8
-
8
-
8
16
H
A
N
G
I
N
G
-
B
Y
-
A
-
T
H
R
E
A
D
-
-
-
-
-
--
-
-
-
-
-
-
1
-
7
-
-
7
-
2
7
-
1
-
2
-
9
5
1
4
+
=
46
4+6
=
10
1+0
1
-
1
-
-
1
-
7
-
-
7
-
2
25
-
1
-
20
-
18
5
1
4
+
=
91
9+1
=
10
1+0
1
-
1
16
H
A
N
G
I
N
G
-
B
Y
-
A
-
T
H
R
E
A
D
-
-
-
-
-
--
-
-
-
-
-
8
1
14
7
9
14
7
-
2
25
-
1
-
20
8
18
5
1
4
+
=
144
1+4+4
=
9
-
9
-
9
-
8
1
5
7
9
5
7
-
2
7
-
1
-
2
8
9
5
1
4
+
=
81
8+1
=
9
1+0
9
-
9
16
H
A
N
G
I
N
G
-
B
Y
-
A
-
T
H
R
E
A
D
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
-
-
-
1
occurs
x
3
=
3
-
3
-
-
-
-
-
-
-
-
-
2
-
-
-
-
2
-
-
-
-
-
-
-
2
occurs
x
2
=
4
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
4
-
-
-
5
-
-
5
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
7
-
-
7
-
-
7
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
3
=
21
2+1
3
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
8
occurs
x
2
=
16
1+6
7
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
16
H
A
N
G
I
N
G
-
B
Y
-
A
-
T
H
R
E
A
D
-
-
36
-
-
16
-
81
-
36
1+6
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
3+6
-
-
1+6
-
8+1
-
3+6
7
H
A
N
G
I
N
G
-
B
Y
-
A
-
T
H
R
E
A
D
-
-
9
-
-
7
-
9
-
9
-
8
1
5
7
9
5
7
-
2
7
-
1
-
2
8
9
5
1
4
-
-
-
-
-
-
-
-
-
-
7
H
A
N
G
I
N
G
-
B
Y
-
A
-
T
H
R
E
A
D
-
-
9
-
-
7
-
9
-
9

 

 

L
=
3
4
LIFE
32
23
5
H
=
8
7
HANGING
60
42
6
B
=
2
2
BY
27
9
9
A
=
1
1
A
1
1
1
T
=
2
6
THREAD
56
29
2
-
-
16
20
First Total
176
104
23
-
-
1+6
2+0
Add to Reduce
1+7+6
1+0+4
2+3
Q
-
7
2
Second Total
14
5
5
-
-
-
-
Reduce to Deduce
1+4
-
-
-
-
7
2
Essence of Number
5
5
5

 

 

-
THREAD
-
-
-
1
R
18
9
9
5
DEATH
38
20
2
6
THREAD
56
29
2

 

 

T
=
2
3
THE
33
15
6
D
=
4
5
DEATH
38
20
2
W
=
5
4
WORD
60
24
6
-
-
11
12
-
131
59
14
T
=
2
3
THE
33
15
6
R
=
9
1
R
18
9
9
I
=
9
2
IN
23
14
5
T
=
2
6
THREAD
56
29
2
-
-
22
12
-
130
67
22
-
-
45
24
Add to Reduce
261
126
36
-
-
4+5
2+4
Reduce to Deduce
2+6+1
1+2+6
3+6
-
-
9
6
Essence of Number
9
9
9

 

LIFE HANGING BY A THREAD

 

-
20
L
I
F
E
-
H
A
N
G
I
N
G
-
B
Y
-
A
-
T
H
R
E
A
D
-
-
-
-
-
--
-
-
-
-
44
-
-
9
-
-
-
8
-
5
-
9
5
-
-
-
-
-
-
-
-
8
-
-
-
-
+
=
44
4+4
=
8
-
8
-
8
62
-
-
9
-
-
-
8
-
14
-
9
14
-
-
-
-
-
-
-
-
8
-
-
-
-
+
=
62
6+2
=
8
-
8
-
8
-
20
L
I
F
E
-
H
A
N
G
I
N
G
-
B
Y
-
A
-
T
H
R
E
A
D
-
-
-
-
-
--
-
-
-
-
60
-
3
-
6
5
-
-
1
-
7
-
-
7
-
2
7
-
1
-
2
-
9
5
1
4
+
=
60
6+0
=
6
-
6
-
6
114
-
12
-
6
5
-
-
1
-
7
-
-
7
-
2
25
-
1
-
20
-
18
5
1
4
+
=
114
1+1+4
=
6
-
6
-
6
-
20
L
I
F
E
-
H
A
N
G
I
N
G
-
B
Y
-
A
-
T
H
R
E
A
D
-
-
-
-
-
--
-
-
-
-
176
-
12
9
6
5
-
8
1
14
7
9
14
7
-
2
25
-
1
-
20
8
18
5
1
4
+
=
176
1+7+6
=
14
1+4
5
-
5
104
-
3
9
6
5
-
8
1
5
7
9
5
7
-
2
7
-
1
-
2
8
9
5
1
4
+
=
104
1+0+4
=
5
1+0
5
-
5
-
20
L
I
F
E
-
H
A
N
G
I
N
G
-
B
Y
-
A
-
T
H
R
E
A
D
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
-
-
-
1
occurs
x
3
=
3
-
3
2
``-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
2
-
-
-
-
-
-
-
2
occurs
x
2
=
4
-
4
3
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
-
3
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
4
5
-
-
-
-
5
-
-
-
5
-
-
5
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
5
occurs
x
4
=
20
2+0
2
6
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
-
6
7
-
-
-
-
-
-
-
-
-
7
-
-
7
-
-
7
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
3
=
21
2+1
3
8
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
8
occurs
x
2
=
16
1+6
7
9
-
-
9
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
9
occurs
x
3
=
27
2+7
9
45
20
L
I
F
E
-
H
A
N
G
I
N
G
-
B
Y
-
A
-
T
H
R
E
A
D
-
-
45
-
-
20
-
104
-
41
4+5
2+0
-
9
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
4+5
-
-
2+0
-
1+0+4
-
4+1
9
2
L
I
F
E
-
H
A
N
G
I
N
G
-
B
Y
-
A
-
T
H
R
E
A
D
-
-
9
-
-
2
-
5
-
5
-
-
3
9
6
5
-
8
1
5
7
9
5
7
-
2
7
-
1
-
2
8
9
5
1
4
-
-
-
-
-
-
-
-
-
-
9
2
L
I
F
E
-
H
A
N
G
I
N
G
-
B
Y
-
A
-
T
H
R
E
A
D
-
-
9
-
-
2
-
5
-
5

 

 

ZERO

ONE TWO THREE FOUR

FIVE

SIX SEVEN EIGHT NINE

 

 

THE DEAD SEA SCROLLS

J. M. Allegro 1956

CHAPTER NINE


THE DOCTRINES OF THE SECT

Page 124

THE basic philosophical and re us conception of the Sect is contained in their doctrine of the Two Spirits. Briefly this implies that there are in the Univese two spirits, one of good and the other of evil, respectively symbolized. as Light and Darkness. Both are under the same supreme rule of God who will eventually give the victory to Good, but only after a prolonged cosmic battle. The war of the Spirits is reflected on earth in the tensions within every man for good and evil, as the. Manual says;

And He assigned to Man two Spirits in which he should until the time of His visitation, They are the spirits of Truth and Perversity: Truth born out of the spring of Light, Perversity from the well of Darkness. The dominion of all the children of righteousness is in the
hands of the Prince of Lights so:that they walk in the ways of Light, whereas the government of the children of Perversity is in the hands of.he Angel of Darkness, to walk in the ways of Darkness. The purpose of the Angel of Darkness is to lead all the children of righteousness astray, and all, their sin, their iniquities, their guilt and - their rebellious works are the result of his domination, in
accordance with. God's mysteries until His appointed time. And all their stripes and seasons of affliction are conseqent upon the rule of his (Satan's) hostility.

Thus the whole cosmos is divided for the time being into two camps, and as Man is apportioned these two spirits so will he behave:

Until now the Spirits of Truth and Perversity struggle within the heart of Man, behaving with wisdom and folly. And according as a man inherits truth and righteousness; so will he hate, Perversion, but in so far as his / Page 125 / heritage is rather from the side of perversion and wickedness, so shall he loathe the Truth.

Another docunient tells us that his 'inheriting' of these Spirits depends on the stars at his birth, and even that the proportions within a man can be numerically reckoned.
:Here are the fruits of the Spirit of Truth as enumerated in the Manual:

To enlighten the heart of Man and to make straight before him all the ways of true righteousness, to make his heart fearful for the judgements of God; a humble spirit, an even temper, a freely compassionate nature, an eternal goodness, and understanding and insight and mighty wisdom which believes in all God's works, and a confident trust in His many mercies, and a spirit of knowledge in every ordered work, and zeal for righteous judgements, and a determined holiness with steadfast mind; loyal feelings towards all the children of Truth, and a radiant purity which loathes every impure idol; a, humble bearing and a discretion regarding all the hidden things of Truth and secrets of Knowledge.

The reward to those who show these qualities in their lives

is healing and abundant peace, length of life and fruitful seed with everlasting blessings, and eternal joy in immortality, a crown of glory and a robe of majesty in eternal light.

To be contrasted with this sublime state is the lot of those led by the Spirit of Perversion. Among the fruits of their Spirit are greed, injustice, wickedness, falsehood, pride, deceit, hasty temper, jealousy, lechery, blasphemy, spiritual obtuseness, and obstinacy, and vile cunning. No wonder that the best he can expect hi the 'Day of Visitation' is many stripes from the Angels of Destruction, in the everlasting Pit, thrtaigh the overwhelming God of Vengeance, in everlasting terror and perpetual disgrace, with the shame of extermination in the Fire of the dark regions. And all their times for all generations will be in grievous mourning and bitter misfortune, in the dark calamities until they are destroyed with no chance to escape.

Page 126

Since the the Spirits are apportioned at birth, this apparent determinism may seem to override the bounds of justice. If a man by his stars, is given a balance of evil in his character it seems hardly fair to condemn him to such punishment for eternity. The argument will have a familiar ring in these days of popular psychology, but the Qumran Covenanter, at least had his answer, For all Men there was one way of salvation depending on his own will and the mercy of God. If he could but apply hirnself to the study of God's Word in humility and pious devotion, God would answer by granting him a restored cleanliness, a sense of perfection.

For it is . . . through the submission of his soul to the statutes of God that his flesh may be cleansed ('flesh being here exactly the Pauline sarrx, the debased moral nature of Man) . . .will order his steps in the Perfect Way and in.all the paths of God not transgressing a single one of His words.

Man must prepare himself by self-discipline, but the action of cleansing is entirely dependent on the will of God. Man has no claim. to justification merely on the grounds of his good works; it ii'an act of divine grace, as much in the eyes of the Covenanter as of Paul.

As for me [says the psalmist at the, end of the Manual], my justification belongs to God, and in His hand is the perfection of my way . . . . and from the fountain of His righteousness (springs) my justification, a light in my heart.

And again,

If I totter, the covenant love of God is my eternal salvation, and if I stumble in the crookedness of my flesh, my justification depends on the righteousness of God, which is eternal.

The word used here for 'justification' is mishpat, which also means 'judgement!'. Man's, justification is the pronounced verdict of God, a .legal 'clearing' which by no means implies sinlessness. Rather, Man's iniquity has been cleansed by the grace of God: he is restored into true sonship / Page 127 / and, in the words of another passage of the Manuel, 'estimated perfect'

In all this, many of my Christian readers will have begun to feel the warmth of a familiar hearth. Here are the ideas of the New Covenant, the emphasis on justification by grace and a doctrine of perfection. We are indeed bordering very closely on to Christian soil and must accordingly begin to, weave our threads of Qumran theology into the fabric of the New Testament to understand fully the considerable significance of the new material for the history of the Church.

Let us first return to the basic doctrine of the Two Spirits. The richest source of New Testament comparison is certainly in the writings of St John. In his first Epistle there is hardly a paragraph which does not contain some reference to the opposition of Light and Darkness, of Truth and Error (a legitimate translation of 'awon, 'perversion', at root, anything twisted).

God is Light, and in Him is no darkness at all. If we say that we have fellowship with Him, and walk in darkness, we lie, and do not the truth (a favourite. Qumran phrase) : but if we walk in the Light, as He is in the Light, we have fellowship one with another (i. 5-7).

The spirits of this world must be tested and proved according to their response to the central fact of creation, the Messiahship of Jesus :

Beloved, believe not every spirit, but prove the spirits, whether they are of God: because many false prophets are gone out into the world . . . Hereby know ye the Spirit of God: every spirit which confesseth that Jesus Christ is come in the flesh is of God: and every spirit which confesseth not Jesus is not of God (iv. 1-3).

Perhaps most familiar is the Prologue of the Gospel:

In him was life; and the life was the Light of men. And the Light shineth in the darkness; and the darkness apprehendeth it not ... There was the true Light, even the Light which lighteth every man coming into the world (i. 4-5, 9).

Page 128

It is a fact that the Qumran library has profoundly affected the study of the Johannine writings and many long- held conceptions have had to be radically revised. No longer can John be regarded as the most Hellenistic of the Evangelists; his `gnosticism' and the whole framework of his thought is seen now to spring directly from a Jewish sectarianism rooted in Palestinian soil, and his material recognized as founded in the earliest layers of Gospel traditions.

In 'the Light which lighteth every man' we have explicitly the idea of apportionment of the Spirit of Light to Man at birth, and perhaps the enigmatic phrase in iii. 34,

for he giveth not the Spirit by measure

has reference to the numerical division of Qumran. To John, the apportionment of the Spirit of Light to Jesus was such that he became Light itself: I am the Light of the world', and he records that the promise to those about him who would believe on him and his mission was that they should become 'sons of Light', the exact terminology used by the Sect to. describe themselves in the apocalyptic war with the 'sons of Darkness '. Jesus speaks of a 'second birth' when a Man would 'be born of the water and the Spirit', and we might recall the Qumran psalm which speaks of God purifying

some of the sons of man to abolish the spirit of perversion from his flesh, and to cleanse him by His Holy Spirit from all wicked deeds, and sprinkle on him the Spirit of Truth as purifying water.

Just as the Qumran sectarians waited for the final vindication of the Spirit of Light at the Time of Visitation, so to John, in a different perspective,

the darkness is passing away, and the true Light already shineth (I. ii. 8).

This opposition of Light and Darkness, Truth and Error, comes clearly enough from Iranian thought, but it did not develop into an absolute dualism at Qumran as it did there. Both good and bad spirits are subject to God, although ,/ Page 129 / naturally enough here, as in John, we are coming perilously near to a dualism in the personification of the Spirit of Evil in the Angel of Darkness, or Belial for Qumran, and Satan, the Devil, the Prince of this world', 'murderer from the beginning', for John. Demon possession is a necessary corollary of this doctrine, and of course occurs time and time again in the gospel stories, particularly in the healing miracles. Jesus used his authority as one abundantly `possessed' of the opposite Spirit, to cast out the powers of darkness in the mentally sick. Thus his enemies' assertion that he was the Devil himself was quite absurd:

and if Satan casteth out Satan, he is divided against himself . . . But if I by the Spirit of God cast out devils, then is the kingdom of God come upon you (Matt. xii. 26-8).

If Jesus is demonstrating the power of the Spirit of Light in this way against the powers of Darkness, it can only mean that the cosmic battle is nearing its climax in the universe, and the `rule' or ' kingdom' of God is being wonderfully demonstrated in the world. God has at last come to the aid of a divided mankind, in the person of His Messiah, or Prince of Light, who enters the house of Satan, the strong man', and despoils it The moral issues of the world take on their true colours : no longer do the greys and half-whites plague man's decisions, but he is confronted with blacks and whites, and the choice is clear-cut :

He that is not with me is against me (Matt. xii. 3o).

To be kept contantly in mind when reading Qumran literature, as also the New Testament, is the sense of - impending doom which pervaded religious thought of this time, and which at intervals, has done so ever since. We have already seen that the Qumran sectarians went into the desert to prepare for the Day of Visitation, and from there they viewed the terrible events in their land and read them as the 'signs of the times '. Jesus, too, is aware of, a special tension in the world, coming to a climax as he faced his death, in which the Spirits of Darkness would make their /Page 130/ final bid for supremacy, but which would in its victory, usher in the new age.This time of trial would. be shared by all living in those days, for in every man the forces of evil would increase.their struggle against the powers of Light and Truth as the end drew near. It was a time of Temptation (peirasmos of the New Testament), and Jesus' hope for his followers was that they should be spared this terrible conflict within their hearts which he himself was undergoing as representative of mankind.

'Pray that ye enter not into temptation is the keynote of his last messages, and when the climax was-drawmg near, and the forces of Darkness drew themselves together for the supreme battle, he bade.his disciples keep awake in the Garden: 'Watch and pray, that ye enter not into temptation.' His pattern of prayer again sounds this note of urgency, though over-repetition would seem to have blunted
most of us. 'Thy kingdom come' is no vague hope for the morrow, but a cry of anguish from the bottom of a tortured soul for the end of the Age, a release from the spiritual battle which the new age of Light and goodness would bring.

'Lead us not into temptation but deliver us from evil' is the plea of a soul battling within itself as the powers of darkness begin to pit their strength against an awkened conscience.

The Mysteries

God, through the.mysteries of His understanding and his glorious wisdom and His glorious wisdom, has ordained a set period for
Perversioni, and in the time of His visitation He will destroy it for ever. Then shall the truth of the tiniverse s hine forth for all time.

Thus speaks the Manual of Discipline; now listen to Paul in his letter to the Romans:

according to the revelation of the rnysstery which hath been kept in silence through times eternal, but now is manifested (xvi. 25-26).

Page 131

And again to the Corinthians:

but we speak God's Wisdom in a Mystery, even the wisdom that hath been hidden, which God foredained before the worlds unto our glory: which none of the rulers of this world knoweth: for had they known it they would not have crucified the Lord of glory . . . But unto us God revealed it through the Spirit: for the Spirit. searcheth, all things, yea, the deep things of God (I. ii. 7.7-10).

And speaking to the.Ephesianst
. . . how that by revelation was made known unto me the thystery, as I wrote afore a few words, whereby, when ye read, ye can perceive my understanding in the mystery of Christ; which in other generations was not made known unto the sons of men, as it hath now been revealed unto his holy apostles and prophets in the spirit. . . ; unto me . . .was this grace given, to preach unto the Gentiles the unsearchable riches of Christ: and to make all men see what is the dispensation of the mystery which from all ages hath been hid in God who created, all things (iii. 3-9).

So possession of the Holy Spirit was to Paul a means of unlocking these divine 'mysteries'. The !teacher of Righteousness of the Qumran. Community also had. access to these secrets, as we learn from the commentary on Habakkuk:

to whom God made known all the secrets of the words of His servants; the prophets.

The 'mystery ' theme, originally traceable to Persian thought and found to some extent even in orthodox Judaism, is very common in Qumran literature, and again there can be little:doubt that Paul was standing in a direct line of tradition with our Sectarians when he used the idea and, indeed, at times the exact terminology, of the Scrolls in this connexion. But, as in his doctrine of justification and
redemption, Paul is looking primarily to the work. and person Jesus, as the, source of grace and knowledge of the mysteries. For Paul, this process of revelation, making the unknown .God 'knowable', was now available not only to / Page 132 / God's servants the prophets' but even to the Gentiles through the universality of the Messiah.

 

 

ANUBIS A NUMBER IS

 

 

DIVINE FEMININE DIVINE MASCULINE DIVINE

DIVINE MASCULINE DIVINE FEMININE DIVINE

 

 

I HAVE COME HAVE I

FRATERNAL GREETINGS AND GREETINGS FRATERNAL

OF

PEACE AND LOVE AND LOVE AND PEACE

ALWAYS

UNTO

ALL SENTIENT BEINGS ALL BEINGS SENTIENT ALL

 

 

27 Aug 2007 ... Frank Drake sat down with Astrobiology Magazine’s Leslie Mullen to .... The price of SETI is not a lot, only a few million dollars a year. .

.astrobio.net/news/print.php?sid=2441

27 Aug 2007 ... at Cornell University and the University of California, Santa Cruz. ... Frank Drake sat down with Astrobiology Magazine’s Leslie Mullen ...

The Man to Contact

In the field of astrobiology, few people have had a bigger influence than Frank Drake. In 1960, he conducted the first radio Search for Extraterrestrial Intelligence (SETI). He formulated the “Drake Equation,” which set the standard for the search for alien life in our galaxy, providing scientific rigor to a field of inquiry that previously had been derided as pure science fiction.

Drake, along with Carl Sagan, designed plaques that were carried on the Pioneer 10 and Pioneer 11 spacecraft. The Pioneer plaques depicted symbolic messages for any aliens the spacecraft might encounter as they travel outside our solar system. Drake also worked with Sagan on theVoyager Golden Record. Containing sounds and images of life on Earth, the record was sent on both the Voyager 1 and Voyager 2 spacecraft."

AM: Listening at the right time, at the right star that has a planet with life at the same point of evolution as us – the chance of that seems so small.


FD: Small, yes, but we live in a galaxy very rich in stars suitable for life. My estimate is we’ll find existing intelligent life in one in ten million stars.

AM: Speaking of listening at the right time, what are your thoughts about the “Wow signal ” received by the Big Ear telescope at Ohio State University in 1977?

FD: It’s an unsolved mystery. It could have been an alien signal, or it could have been a human signal inadvertently picked up, or something else, perhaps an equipment failure. Hundreds of people have looked for that signal over the years, but it’s never been repeated.

There have been some other tantalizing candidate signals. The long Harvard search of Horowitz and Sagan observed more than thirty signals that had the earmark of an extraterrestrial signal. The SETI@Home program has observed more than a hundred such signals. Both of these programs are automated, though, so no one was there at the time to do immediate follow-up observations. Researchers later tried to detect these signals, but, as with the Wow signal, they’ve been unsuccessful. So the origin of these signals is an open question.

Project Phoenix of the SETI Institute also has found many good candidates, but that program could immediately determine the origin of the signal and all of them turned out to be of human origin. It may be that all the potential signals detected so far were generated by humans. But for now they remain a mystery, and that gives hope to those of us who search for alien signals."

Extracts posted 27/8/08

 

 

DAILY MAIL

Thursday, September 11, 2008

Pages 12/13

"BANG! Day the/world didn't end"

Page 12

'Secrets of the universe' machine is turned on. . . but we're till here

Michael Hanlon Science Editor

Page 12/13

"Suffering superlatives/or how Marr got his particles all shook up"

Page 13

"A few of them said 'wow! from time to time but there was nothing much to see or hear"

Quentin Letts

 

 

HOLY BIBLE

C

9 V 9

Page 575

JOB

Which maketh Arcturus, Orion, and Pleiades, and the chambers of the south.

 

 

THE

MEASURERS

MAGIMAGICMAGI

SEE

 

 

GOD IS ALIVE
MAGIC IS AFOOT

words: Leonard Cohen music: Buffy Sainte-Marie

1

God is alive; Magic is afoot
God is alive; Magic is afoot
God is alive; Magic is afoot
Alive is afoot.....
Magic never died.

God never sickened;
many poor men lied
many sick men lied
Magic never weakened
Magic never hid
Magic always ruled
God is afoot
God never died.

God was ruler
though his funeral lengthened
Though his mourners thickened
Magic never fled
Though his shrouds were hoisted
the naked God did live
Though his words were twisted
the naked Magic thrived
Though his death was published
round and round the world
the heart did not believe

Many hurt men wondered
many struck men bled
Magic never faltered
Magic always led.
Many stones were rolled
but God would not lie down
Many wild men lied
many fat men listened
Though they offered stones
Magic still was fed
Though they locked their coffers
God was always served.

2

Magic is afoot. God rules.
Alive is afoot. Alive is in command.
Many weak men hungered
Many strong men thrived
Though they boasted solitude
God was at their side
Nor the dreamer in his cell
nor the captain on the hill
Magic is alive
Though his death was pardoned
round and round the world
the heart did not believe.

Though laws were carved in marble
they could not shelter men
Though altars built in parliaments
they could not order men
Police arrested Magic
and Magic went with them,
for Magic loves the hungry.

But Magic would not tarry
it moves from arm to arm
it would not stay with them
Magic is afoot
it cannot come to harm
it rests in an empty palm
it spawns in an empty mind
but Magic is no instrument
Magic is the end.

Many men drove Magic
but Magic stayed behind
Many strong men lied
they only passed through Magic
and out the other side
Many weak men lied
they came to God in secret
and though they left him nourished
they would not say who healed
Though mountains danced before them
they said that God was dead
Though his shrouds were hoisted
the naked God did live

3

This I mean to whisper to my mind
This I mean to laugh with in my mind
This I mean my mind to serve 'til
service is but Magic
moving through the world
and mind itself is Magic
coursing through the flesh
and flesh itself is Magic
dancing on a clock
and time itself the magic length of God.

 

 

SURE I DREAM AS THE HAMMER STRIKES THE ANVIL

AND I DREAM AS THE SPARKS FALL ON THE FLOOR

 

 

3
THE
33
15
6
4
MIND
40
22
4
2
OF
21
12
3
9
HUMANKIND
95
41
5
18
First Total
189
90
18
1+8
Add to Reduce
1+8+9
9+0
1+8
9
Second Total
18
9
9
-
Reduce to Deduce
1+8
-
-
9
Essence of Number
9
9
9

 

 

3
THE
33
15
6
7
SUPREME
97
34
7
4
SELF
42
15
6
14
First Total
172
64
19
1+4
Add to Reduce
1+7+2
6+4
1+9
5
Second Total
10
10
10
-
Reduce to Deduce
1+0
1+0
1+0
5
Essence of Number
1
1
1

 

 

THE

SUPREME SELF SUPREME

LIVING FORM FROM OUT THE IN OF GODS LIVING FORM

 

 

I

ME

YOU

ENTANGLEMENT

MENTAL ANGLE ANGEL ANGLE MENTAL

ENTANGLES MENTALLY MENTALLY ENTANGLES

KARMAS THOUGHT ENTANGLEMENT THOUGHT KARMAS

123456789 REAL REALITY REVEALED REALITY REAL 987654321

 

 

ONE I ONE ME ONE YOU ONE

GOD

UNIVERSAL

MIND

INTO THE WITHIN OF THE WITHOUT THAT

ISISISISISIS

THOUGHT DIVINE THOUGHT

111111111x111111111

12345678987654321

ABCDEFGHIJKLMNOPQRSTUVWXYZ WELCOME HERO WELCOME ABCDEFGHIJKLMNOPQRSTUVWXYZ

 

 

ADVENT 887 ADVENT

 

3
THE
33
15
6
3
LAW
36
9
9
2
OF
21
12
3
6
KARMAS
63
18
9
14
Add to Reduce
153
54
27
1+4
Reduce to Deduce
1+5+3
5+4
2+7
5
Essence of Number
9
9
9

 

 

3
THE
33
15
6
3
LAW
36
9
9
2
OF
21
12
3
5
CAUSE
49
13
4
3
AND
19
10
1
6
EFFECT
45
27
9
22
First Total
203
86
32
2+2
Add to Reduce
2+0+3
8+6
3+2
4
Second Total
5
14
5
-
Reduce to Deduce
-
1+4
-
4
Essence of Number
5
5
5

 

 

21-
22
T
H
E
-
L
A
W
-
O
F
-
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
6
-
-
-
-
-
1
-
-
-
5
-
-
-
-
-
-
-
-
+
=
20
2+0
=
2
=
2
=
2
-
-
-
8
-
-
-
-
-
-
15
-
-
-
-
-
19
-
-
-
14
-
-
-
-
-
-
-
-
+
=
56
5+6
=
11
1+1
2
=
2
-
22
T
H
E
-
L
A
W
-
O
F
-
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
-
-
-
-
-
-
-
-
-
-
2
-
5
-
3
1
5
-
-
6
-
3
1
3
-
5
-
1
-
4
-
5
6
6
5
3
2
+
=
66
6+6
=
12
1+2
3
=
3
-
-
20
-
5
-
12
1
23
-
-
6
-
3
1
21
-
5
-
1
-
4
-
5
6
6
5
3
20
+
=
147
1+4+7
=
12
1+2
3
=
3
-
22
T
H
E
-
L
A
W
-
O
F
-
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
-
-
-
--
-
-
-
-
-
-
20
8
5
-
12
1
23
-
15
6
-
3
1
21
19
5
-
1
14
4
-
5
6
6
5
3
20
+
=
203
2+0+3
=
5
=
5
=
5
-
-
2
8
5
-
3
1
5
-
6
6
-
3
1
3
1
5
-
1
5
4
-
5
6
6
5
3
2
+
=
86
8+6
=
14
1+4
5
=
5
-
22
T
H
E
-
L
A
W
-
O
F
-
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
-
1
-
-
1
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
4
=
4
=
4
-
``-
2
-
-
-`
-
-
-
-`
-
-
-`
-
-
-
-
-
-`
-
-
-
-`
-
-
-
-
-
2
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
3
-
-
-
-
-
-
3
-
3
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
3
occurs
x
4
=
12
1+2
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
5
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
5
-
-
5
-
-
5
-
-
-
-
5
occurs
x
6
=
30
3+0
3
-
-
-
-
-
-
-
-
-
-
6
6
-
-
-
-
-
-
-
-
-
-
-
-
6
6
-
-
-
-
-
6
occurs
x
4
=
24
2+4
6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
16
22
T
H
E
-
L
A
W
-
O
F
-
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
29
-
-
22
-
86
-
32
1+6
2+2
-
-
5
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
5
-
-
5
-
-
5
-
-
-
-
2+9
-
-
2+2
-
8+6
-
3+2
7
4
T
H
E
-
L
A
W
-
O
F
-
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
11
-
-
4
-
14
-
5
-
-
2
8
5
-
3
1
5
-
6
6
-
3
1
3
1
5
-
1
5
4
-
5
6
6
5
3
2
-
-
1+1
-
-
-
-
1+4
-
-
7
4
T
H
E
-
L
A
W
-
O
F
-
C
A
U
S
E
-
A
N
D
-
E
F
F
E
C
T
-
-
2
-
-
4
-
5
-
5

 

 

WHY SMASH ATOMS

A. K. Solomon 1940

Page 77

"Once the fairy tale hero has penetrated the ring of fire round the magic mountain he is free to woo the heroine in her castle on the mountain top."

 

 

3
THE
33
15
6
4
RING
48
30
3
2
OF
21
12
3
4
FIRE
38
29
2
13
First Total
140
86
14
1+3
Add to Reduce
1+4+0
8+6
1+4
4
Second Total
5
5
5
-
Reduce to Deduce
-
1+4
-
4
Essence of Number
5
5
5

 

 

-
13
T
H
E
-
R
I
N
G
-
O
F
-
F
I
R
E
-
-
-
-
-
--
-
-
-
-
-
-
-
8
-
-
-
9
5
-
-
6
-
-
-
9
-
-
+
=
37
3+7
=
10
1+0
1
=
1
-
-
-
8
-
-
-
9
14
-
-
15
-
-
-
9
-
-
+
=
55
5+5
=
10
1+0
1
=
1
-
13
T
H
E
-
R
I
N
G
-
O
F
-
F
I
R
E
-
-
-
-
-
-
-
-
-
-
-
-
2
-
5
-
9
-
-
7
-
-
6
-
6
-
9
5
+
=
49
4+9
=
13
1+3
4
=
4
-
-
20
-
5
-
18
-
-
7
-
-
6
-
6
-
18
5
+
=
85
8+5
=
13
1+3
4
=
4
-
13
T
H
E
-
R
I
N
G
-
O
F
-
F
I
R
E
-
-
-
-
-
--
-
-
-
-
-
-
20
8
5
-
18
9
14
7
-
15
6
-
6
9
18
5
+
=
140
1+4+0
=
5
=
5
=
5
-
-
2
8
5
-
9
9
5
7
-
6
6
-
6
9
9
5
+
=
86
8+6
=
14
1+4
5
=
5
-
13
T
H
E
-
R
I
N
G
-
O
F
-
F
I
R
E
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
5
-
-
-
--
-
--
-
-
5
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
-
-
-
-
-
6
6
-
6
-
-
-
-
-
6
occurs
x
3
=
18
1+8
9
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
-
-
9
9
-
-
-
-
-
-
-
9
9
-
-
-
9
occurs
x
4
=
36
3+6
9
8
13
T
H
E
-
R
I
N
G
-
O
F
-
F
I
R
E
-
-
37
-
-
13
-
86
-
41
-
1+3
-
-
-
-
9
9
-
-
-
-
--
-
--
9
9
-
-
-
3+7
-
-
1+3
-
8+6
-
4+1
8
4
T
H
E
-
R
I
N
G
-
O
F
-
F
I
R
E
-
-
10
-
-
4
-
14
-
5
-
-
2
8
5
-
9
9
5
7
-
6
6
-
6
9
9
5
-
-
1+0
-
-
-
-
1+4
-
-
8
4
T
H
E
-
R
I
N
G
-
O
F
-
F
I
R
E
-
-
1
-
-
4
-
5
-
5

 

 

3
T
H
E
-
R
I
N
G
-
O
F
-
F
I
R
E
-
-
-
-
-
--
-
-
-
-
-
-
8
-
-
-
9
5
-
-
6
-
-
-
9
-
-
+
=
37
3+7
=
10
1+0
1
=
1
-
-
8
-
-
-
9
14
-
-
15
-
-
-
9
-
-
+
=
55
5+5
=
10
1+0
1
=
1
13
T
H
E
-
R
I
N
G
-
O
F
-
F
I
R
E
-
-
-
-
-
-
-
-
-
-
-
2
-
5
-
9
-
-
7
-
-
6
-
6
-
9
5
+
=
49
4+9
=
13
1+3
4
=
4
-
20
-
5
-
18
-
-
7
-
-
6
-
6
-
18
5
+
=
85
8+5
=
13
1+3
4
=
4
13
T
H
E
-
R
I
N
G
-
O
F
-
F
I
R
E
-
-
-
-
-
--
-
-
-
-
-
20
8
5
-
18
9
14
7
-
15
6
-
6
9
18
5
+
=
140
1+4+0
=
5
=
5
=
5
-
2
8
5
-
9
9
5
7
-
6
6
-
6
9
9
5
+
=
86
8+6
=
14
1+4
5
=
5
13
T
H
E
-
R
I
N
G
-
O
F
-
F
I
R
E
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
5
-
-
-
5
-
-
-
--
-
--
-
-
5
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
-
-
-
-
6
6
-
6
-
-
-
-
-
6
occurs
x
3
=
18
1+8
9
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
-
9
9
-
-
-
-
-
-
-
9
9
-
-
-
9
occurs
x
4
=
36
3+6
9
13
T
H
E
-
R
I
N
G
-
O
F
-
F
I
R
E
-
-
37
-
-
13
-
86
-
41
1+3
-
-
-
-
9
9
-
-
-
-
--
-
--
9
9
-
-
-
3+7
-
-
1+3
-
8+6
-
4+1
4
T
H
E
-
R
I
N
G
-
O
F
-
F
I
R
E
-
-
10
-
-
4
-
14
-
5
-
2
8
5
-
9
9
5
7
-
6
6
-
6
9
9
5
-
-
1+0
-
-
-
-
1+4
-
-
4
T
H
E
-
R
I
N
G
-
O
F
-
F
I
R
E
-
-
1
-
-
4
-
5
-
5

 

 

THE FIERY RING FIERY THE RING

 

3
THE
33
15
6
5
FIERY
63
36
9
4
RING
48
30
3
12
Add to Reduce
144
81
18
1+2
Reduce to Deduce
1+4+4
8+1
1+8
3
Essence of Number
9
9
9

 

 

-
CIRCLE
-
-
-
1
C
3
3
3
1
I
9
9
9
1
R
18
9
9
1
C
3
3
3
1
L
12
3
3
1
E
5
5
5
6
CIRCLE
50
32
32
-
-
5+0
3+2
3+2
6
CIRCLE
5
5
5

 

 

-
CIRCLE
-
-
-
5
CIRCL
45
27
9
1
E
5
5
5
6
CIRCLE
50
32
32
-
-
5+0
3+2
3+2
6
CIRCLE
5
5
5

 

 

-
6
C
I
R
C
L
E
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
`-
+
=
9
-
=
9
=
9
=
9
-
-
-
9
-
-
-
`-
+
=
9
-
=
9
=
9
=
9
-
6
C
I
R
C
L
E
-
-
-
-
-
-
-
-
-
-
-
-
3
-
9
3
3
5
+
=
23
2+3
=
5
=
5
=
5
-
-
3
-
18
3
12
5
+
=
32
3+2
=
5
=
5
=
5
-
6
C
I
R
C
L
E
-
-
-
-
-
-
-
-
-
-
-
-
3
9
18
3
12
5
+
=
50
5+0
=
5
=
5
=
5
-
-
3
9
9
3
3
5
+
=
32
3+2
=
5
=
5
=
5
-
6
C
I
R
C
L
E
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
3
-
-
3
3
-
-
-
3
occurs
x
3
=
9
=
9
4
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
=
5
6
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
9
9
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
28
6
C
I
R
C
L
E
-
-
17
-
-
6
-
32
-
23
2+8
-
-
9
9
-
-
-
-
-
1+7
-
-
-
-
3+2
-
2+3
10
6
C
I
R
C
L
E
-
-
8
-
-
6
-
5
-
5
1+0
-
3
9
9
3
3
5
-
-
-
-
-
-
-
-
-
-
1
6
C
I
R
C
L
E
-
-
8
-
-
6
-
5
-
5

 

 

6
C
I
R
C
L
E
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
`-
+
=
9
-
=
9
=
9
=
9
6
C
I
R
C
L
E
-
-
-
-
-
-
-
-
-
-
-
3
-
9
3
3
5
+
=
23
2+3
=
5
=
5
=
5
-
3
-
18
3
12
5
+
=
32
3+2
=
5
=
5
=
5
6
C
I
R
C
L
E
-
-
-
-
-
-
-
-
-
-
-
3
9
18
3
12
5
+
=
50
5+0
=
5
=
5
=
5
-
3
9
9
3
3
5
+
=
32
3+2
=
5
=
5
=
5
6
C
I
R
C
L
E
-
-
-
-
-
-
-
-
-
-
-
3
-
-
3
3
-
-
-
3
occurs
x
3
=
9
=
9
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
=
5
-
-
9
9
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
6
C
I
R
C
L
E
-
-
17
-
-
6
-
32
-
23
-
-
9
9
-
-
-
-
-
1+7
-
-
-
-
3+2
-
2+3
6
C
I
R
C
L
E
-
-
8
-
-
6
-
5
-
5
-
3
9
9
3
3
5
-
-
-
-
-
-
-
-
-
-
6
C
I
R
C
L
E
-
-
8
-
-
6
-
5
-
5

 

 

8
CIRCULAR
85
40
4
6
MOTION
86
32
5
14
Add to Reduce
171
72
9
1+4
Reduce to Deduce
1+7+1
7+2
-
5
Essence of Number
9
9
9

 

 

-
14
C
I
R
C
U
L
A
R
-
M
O
T
I
O
N
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
6
-
9
6
5
+
=
35
3+5
8
-
=
8
=
8
-
-
-
9
-
-
-
-
-
-
-
-
15
-
9
15
14
+
=
62
6+2
8
-
=
8
=
8
-
14
C
I
R
C
U
L
A
R
-
M
O
T
I
O
N
-
-
-
-
-
-
-
-
-
-
-
-
3
-
9
3
3
3
1
9
-
4
-
2
-
-
-
+
=
37
3+7
10
1+0
=
1
=
1
-
-
3
-
18
3
21
12
1
18
-
13
-
20
-
-
-
+
=
109
1+0+9
10
1+0
=
1
=
1
-
14
C
I
R
C
U
L
A
R
-
M
O
T
I
O
N
-
-
-
-
-
-
-
-
-
-
-
-
3
9
18
3
21
12
1
18
-
13
15
20
9
15
14
+
=
171
1+7+1
9
-
=
9
=
9
-
-
3
9
9
3
3
3
1
9
-
4
6
2
9
6
5
+
=
72
7+2
9
-
=
9
=
9
-
14
C
I
R
C
U
L
A
R
-
M
O
T
I
O
N
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
-
2
-
-
3
-
-
3
3
3
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
4
=
12
1+2
3
-
-
-
-
-
-
-
-
1
-
-
4
-
-
-
-
-
-
-
4
occurs
x
1
=
4
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
-
5
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
6
-
-
-
6
occurs
x
2
=
12
1+2
3
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
9
9
-
-
-
-
9
-
-
-
-
9
-
-
-
-
9
occurs
x
4
=
36
3+6
9
15
14
C
I
R
C
U
L
A
R
-
M
O
T
I
O
N
-
-
30
-
-
14
-
72
-
27
1+5
1+4
-
9
9
-
-
-
-
9
-
-
-
-
9
-
-
-
-
3+0
-
-
1+4
-
7+2
-
2+7
6
5
C
I
R
C
U
L
A
R
-
M
O
T
I
O
N
-
-
3
-
-
5
-
9
-
9
-
-
3
9
9
3
3
3
6
9
-
4
6
6
9
6
5
-
-
-
-
-
-
  
-
-
-
6
5
C
I
R
C
U
L
A
R
-
M
O
T
I
O
N
-
-
3
-
-
5
-
9
-
9

 

 

-
14
C
I
R
C
U
L
A
R
-
M
O
T
I
O
N
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
6
-
9
6
5
+
=
35
3+5
8
-
=
8
=
8
-
-
-
9
-
-
-
-
-
-
-
-
15
-
9
15
14
+
=
62
6+2
8
-
=
8
=
8
-
14
C
I
R
C
U
L
A
R
-
M
O
T
I
O
N
-
-
-
-
-
-
-
-
-
-
-
-
3
-
9
3
3
3
1
9
-
4
-
2
-
-
-
+
=
37
3+7
10
1+0
=
1
=
1
-
-
3
-
18
3
21
12
1
18
-
13
-
20
-
-
-
+
=
109
1+0+9
10
1+0
=
1
=
1
-
14
C
I
R
C
U
L
A
R
-
M
O
T
I
O
N
-
-
-
-
-
-
-
-
-
-
-
-
3
9
18
3
21
12
1
18
-
13
15
20
9
15
14
+
=
171
1+7+1
9
-
=
9
=
9
-
-
3
9
9
3
3
3
1
9
-
4
6
2
9
6
5
+
=
72
7+2
9
-
=
9
=
9
-
14
C
I
R
C
U
L
A
R
-
M
O
T
I
O
N
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
-
2
-
-
3
-
-
3
3
3
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
4
=
12
1+2
3
-
-
-
-
-
-
-
-
1
-
-
4
-
-
-
-
-
-
-
4
occurs
x
1
=
4
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
1
=
5
-
5
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
6
-
-
-
6
occurs
x
2
=
12
1+2
3
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
9
9
-
-
-
-
9
-
-
-
-
9
-
-
-
-
9
occurs
x
4
=
36
3+6
9
15
14
C
I
R
C
U
L
A
R
-
M
O
T
I
O
N
-
-
30
-
-
14
-
72
-
27
1+5
1+4
-
9
9
-
-
-
-
9
-
-
-
-
9
-
-
-
-
3+0
-
-
1+4
-
7+2
-
2+7
6
5
C
I
R
C
U
L
A
R
-
M
O
T
I
O
N
-
-
3
-
-
5
-
9
-
9
-
-
3
9
9
3
3
3
6
9
-
4
6
6
9
6
5
-
-
-
-
-
-
  
-
-
-
6
5
C
I
R
C
U
L
A
R
-
M
O
T
I
O
N
-
-
3
-
-
5
-
9
-
9

 

ROUNDANDROUNDROUNDANDROUNDROUNDANDROUNDROUND ANDROUNDROUNDANDROUND

 

 

-
9
C
L
O
C
K
W
I
S
E
-
-
-
-
-
--
-
-
-
-
-
-
-
-
6
-
-
-
9
1
-
+
=
16
1|+6
=
7
=
7
=
7
-
-
-
-
15
-
-
-
9
19
-
+
=
43
4+3
=
7
=
7
=
7
-
9
C
L
O
C
K
W
I
S
E
-
-
-
-
-
--
-
-
-
-
-
-
3
3
-
3
2
5
-
-
5
+
=
21
2+1
=
3
=
3
=
3
-
-
3
12
-
3
11
23
-
-
5
+
=
57
5+7
=
12
1+2
3
=
3
-
9
C
L
O
C
K
W
I
S
E
-
-
-
-
-
--
-
-
-
-
-
-
3
12
15
3
11
23
9
19
5
+
=
100
1+0+0
=
1
=
1
=
1
-
-
3
3
6
3
2
5
9
1
5
+
=
37
3+7
=
10
1+0
1
=
1
-
9
C
L
O
C
K
W
I
S
E
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
occurs
x
1
=
1
=
1
-
``-
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
3
3
-
3
-
-
-
-
-
-
-
3
occurs
x
3
=
9
=
9
4
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
5
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
6
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
1
=
9
=
9
19
9
C
L
O
C
K
W
I
S
E
-
-
26
-
-
9
-
37
-
28
1+9
-
-
-
-
-
-
-
9
-
-
-
-
2+6
-
-
-
-
3+7
-
2+8
10
9
C
L
O
C
K
W
I
S
E
-
-
8
-
-
9
-
10
-
10
1+0
-
3
3
6
3
2
5
9
1
5
-
-
-
-
-
-
-
1+0
-
1+0
1
9
C
L
O
C
K
W
I
S
E
-
-
8
-
-
9
-
1
-
1

 

 

9
C
L
O
C
K
W
I
S
E
-
-
-
-
-
--
-
-
-
-
-
-
-
6
-
-
-
9
1
-
+
=
16
1|+6
=
7
=
7
=
7
-
-
-
15
-
-
-
9
19
-
+
=
43
4+3
=
7
=
7
=
7
9
C
L
O
C
K
W
I
S
E
-
-
-
-
-
--
-
-
-
-
-
3
3
-
3
2
5
-
-
5
+
=
21
2+1
=
3
=
3
=
3
-
3
12
-
3
11
23
-
-
5
+
=
57
5+7
=
12
1+2
3
=
3
9
C
L
O
C
K
W
I
S
E
-
-
-
-
-
--
-
-
-
-
-
3
12
15
3
11
23
9
19
5
+
=
100
1+0+0
=
1
=
1
=
1
-
3
3
6
3
2
5
9
1
5
+
=
37
3+7
=
10
1+0
1
=
1
9
C
L
O
C
K
W
I
S
E
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
occurs
x
1
=
1
=
1
``-
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
3
3
-
3
-
-
-
-
-
-
-
3
occurs
x
3
=
9
=
9
-
-
-
-
-
-
5
-
-
5
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
6
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
1
=
9
=
9
9
C
L
O
C
K
W
I
S
E
-
-
26
-
-
9
-
37
-
28
-
-
-
-
-
-
-
9
-
-
-
-
2+6
-
-
-
-
3+7
-
2+8
9
C
L
O
C
K
W
I
S
E
-
-
8
-
-
9
-
10
-
10
-
3
3
6
3
2
5
9
1
5
-
-
-
-
-
-
-
1+0
-
1+0
9
C
L
O
C
K
W
I
S
E
-
-
8
-
-
9
-
1
-
1

 

 

-
13
A
N
T
I
-
C
L
O
C
K
W
I
S
E
-
-
-
-
-
--
-
-
-
-
-
-
-
5
-
9
-
-
-
6
-
-
-
9
1
-
+
=
30
3+0
=
3
=
3
=
3
-
-
-
14
-
9
-
-
-
15
-
-
-
9
19
-
+
=
66
6+6
=
12
1+2
3
=
3
-
13
A
N
T
I
-
C
L
O
C
K
W
I
S
E
-
-
-
-
-
--
-
-
-
-
-
-
1
-
2
-
-
3
3
-
3
2
5
-
-
5
+
=
24
2+4
=
6
=
6
=
6
-
-
1
-
20
-
-
3
12
-
3
11
23
-
-
5
+
=
78
7+8
=
15
1+5
6
=
6
-
13
A
N
T
I
-
C
L
O
C
K
W
I
S
E
-
-
-
-
-
--
-
-
-
-
-
-
1
14
20
9
-
3
12
15
3
11
23
9
19
5
+
=
144
1+4+4
=
9
=
9
=
9
-
-
1
5
2
9
-
3
3
6
3
2
5
9
1
5
+
=
54
5+4
=
9
=
9
=
9
-
13
A
N
T
I
-
C
L
O
C
K
W
I
S
E
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
2
-
-
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
-
3
3
-
3
-
-
-
-
-
-
-
3
occurs
x
3
=
9
=
9
4
-
-
--
--
--
--
--
--
--
--
--
--
--
--
--
-
-
4
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
5
--
--
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
2
=
18
1+8
9
19
13
A
N
T
I
-
C
L
O
C
K
W
I
S
E
-
-
26
-
-
13
-
54
-
36
1+9
1+3
1
5
2
9
-
3
3
6
3
2
5
9
1
5
-
-
2+6
-
-
1+3
-
5+4
-
3+6
1
4
A
N
T
I
-
C
L
O
C
K
W
I
S
E
-
-
8
-
-
4
-
9
-
9
1+0
-
1
5
2
9
-
3
3
6
3
2
5
9
1
5
-
-
-
-
-
-
-
-
-
-
1
4
A
N
T
I
-
C
L
O
C
K
W
I
S
E
-
-
8
-
-
4
-
9
-
9

 

 

-
13
A
N
T
I
-
C
L
O
C
K
W
I
S
E
-
-
-
-
-
--
-
-
-
-
-
-
-
5
-
9
-
-
-
6
-
-
-
9
1
-
+
=
30
3+0
=
3
=
3
=
3
-
-
-
14
-
9
-
-
-
15
-
-
-
9
19
-
+
=
66
6+6
=
12
1+2
3
=
3
-
13
A
N
T
I
-
C
L
O
C
K
W
I
S
E
-
-
-
-
-
--
-
-
-
-
-
-
1
-
2
-
-
3
3
-
3
2
5
-
-
5
+
=
24
2+4
=
6
=
6
=
6
-
-
1
-
20
-
-
3
12
-
3
11
23
-
-
5
+
=
78
7+8
=
15
1+5
6
=
6
-
13
A
N
T
I
-
C
L
O
C
K
W
I
S
E
-
-
-
-
-
--
-
-
-
-
-
-
1
14
20
9
-
3
12
15
3
11
23
9
19
5
+
=
144
1+4+4
=
9
=
9
=
9
-
-
1
5
2
9
-
3
3
6
3
2
5
9
1
5
+
=
54
5+4
=
9
=
9
=
9
-
13
A
N
T
I
-
C
L
O
C
K
W
I
S
E
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
2
-
-
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
-
3
3
-
3
-
-
-
-
-
-
-
3
occurs
x
3
=
9
=
9
4
-
-
--
--
--
--
--
--
--
--
--
--
--
--
--
-
-
4
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
5
--
--
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
2
=
18
1+8
9
19
13
A
N
T
I
-
C
L
O
C
K
W
I
S
E
-
-
26
-
-
13
-
54
-
36
1+9
1+3
1
5
2
9
-
3
3
6
3
2
5
9
1
5
-
-
2+6
-
-
1+3
-
5+4
-
3+6
1
4
A
N
T
I
-
C
L
O
C
K
W
I
S
E
-
-
8
-
-
4
-
9
-
9
1+0
-
1
5
2
9
-
3
3
6
3
2
5
9
1
5
-
-
-
-
-
-
-
-
-
-
1
4
A
N
T
I
-
C
L
O
C
K
W
I
S
E
-
-
8
-
-
4
-
9
-
9

 

 

-
23
T
H
E
-
E
L
E
C
T
R
O
-
M
A
G
N
E
T
I
C
-
F
I
E
L
D
-
-
-
-
-
--
-
-
-
-
37
-
-
8
-
-
-
-
-
-
-
-
6
-
-
-
-
5
-
-
9
-
-
-
9
-
-
-
+
=
37
3+7
=
10
1+0
1
=
1
55
-
-
8
-
-
-
-
-
-
-
-
15
-
-
-
-
14
-
-
9
-
-
-
9
-
-
-
+
=
55
5+5
=
10
1+0
1
=
1
-
23
T
H
E
-
E
L
E
C
T
R
O
-
M
A
G
N
E
T
I
C
-
F
I
E
L
D
-
-
-
-
-
--
-
-
-
-
74
-
2
-
5
-
5
3
5
3
2
9
-
-
4
1
7
-
5
2
-
3
-
6
-
5
3
4
+
=
74
7+4
=
11
1+1
2
=
2
164
-
20
-
5
-
5
12
5
3
20
18
-
-
13
1
7
-
5
20
-
3
-
6
-
5
12
4
+
=
164
1+6+4
=
11
1+1
2
=
2
-
23
T
H
E
-
E
L
E
C
T
R
O
-
M
A
G
N
E
T
I
C
-
F
I
E
L
D
-
-
-
-
-
--
-
-
-
-
219
-
20
8
5
-
5
12
5
3
20
18
15
-
13
1
7
14
5
20
9
3
-
6
9
5
12
4
+
=
219
2+1+9
=
12
1+2
3
=
3
111
-
2
8
5
-
5
3
5
3
2
9
6
-
4
1
7
5
5
2
9
3
-
6
9
5
3
4
+
=
111
1+1+1
=
3
=
3
=
3
-
23
T
H
E
-
E
L
E
C
T
R
O
-
M
A
G
N
E
T
I
C
-
F
I
E
L
D
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
2
-
2
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
2
occurs
x
3
=
6
=
6
3
-
-
-
-
-
-
3
-
3
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
3
-
-
-
3
occurs
x
4
=
12
1+2
3
4
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
2
=
8
=
8
5
-
-
-
5
-
5
-
5
-
-
-
-
-
-
-
5
-
5
-
-
-
-
-
-
5
-
-
-
-
5
occurs
x
6
=
30
3+0
3
6
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
6
--
-
-
-
-
-
6
occurs
x
2
=
12
1+2
3
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
8
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
9
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
-
9
-
-
-
-
-
9
occurs
x
3
=
27
2+7
9
45
23
T
H
E
-
E
L
E
C
T
R
O
-
M
A
G
N
E
T
I
C
-
F
I
E
L
D
-
-
45
-
-
23
-
111
-
48
4+5
2+3
-
-
5
-
5
-
5
-
-
-
-
-
-
-
5
-
5
-
-
-
-
-
-
5
-
-
-
-
4+5
-
-
2+3
-
1+1+1
-
4+8
9
5
T
H
E
-
E
L
E
C
T
R
O
-
M
A
G
N
E
T
I
C
-
F
I
E
L
D
-
-
9
-
-
5
-
3
-
12
-
-
2
8
5
-
5
3
5
3
2
9
6
-
4
1
7
5
5
2
9
3
-
6
9
5
3
4
-
-
-
-
-
-
-
-
-
1+2
9
5
T
H
E
-
E
L
E
C
T
R
O
-
M
A
G
N
E
T
I
C
-
F
I
E
L
D
-
-
9
-
-
5
-
3
-
3

 

 

-
15
E
L
E
C
T
R
O
M
A
G
N
E
T
I
C
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
5
-
-
9
-
+
=
20
2+0
=
2
=
2
=
2
-
-
-
-
-
-
-
-
15
-
-
-
14
-
-
9
-
+
=
38
3+8
=
11
1+1
2
=
2
-
15
E
L
E
C
T
R
O
M
A
G
N
E
T
I
C
-
-
-
-
-
--
-
-
-
-
-
-
5
3
5
3
2
9
-
4
1
7
-
5
2
-
3
+
=
49
4+9
=
13
1+3
4
=
4
-
-
5
12
5
3
20
18
-
13
1
7
-
5
20
-
3
+
=
112
1+1+2
=
4
=
4
=
4
-
15
E
L
E
C
T
R
O
M
A
G
N
E
T
I
C
-
-
-
-
-
--
-
-
-
-
-
-
5
12
5
3
20
18
15
13
1
7
14
5
20
9
3
+
=
150
1+5+0
=
6
=
6
=
6
-
-
5
3
5
3
2
9
6
4
1
7
5
5
2
9
3
+
=
69
6+9
=
15
1+5
6
=
6
-
15
E
L
E
C
T
R
O
M
A
G
N
E
T
I
C
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
-
-
-
2
-
-
-
-
-
-
-
2
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
3
-
3
-
-
-
-
-
-
-
-
-
-
3
-
-
3
occurs
x
3
=
9
=
9
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
5
-
5
-
-
-
-
-
-
-
5
5
-
-
-
-
-
5
occurs
x
4
=
20
2+0
2
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
9
-
-
-
9
occurs
x
2
=
18
1+8
9
8
15
E
L
E
C
T
R
O
M
A
G
N
E
T
I
C
-
-
37
-
-
15
-
69
-
42
-
-
5
-
5
-
-
-
-
-
-
-
5
5
-
-
-
-
-
3+7
-
-
1+5
-
6+9
-
4+2
-
15
E
L
E
C
T
R
O
M
A
G
N
E
T
I
C
-
-
10
-
-
6
-
15
-
6
-
1+5
5
3
5
3
2
9
6
4
1
7
5
5
2
9
3
-
-
1+0
-
-
-
-
1+5
-
-
-
6
E
L
E
C
T
R
O
M
A
G
N
E
T
I
C
-
-
1
-
-
6
-
6
-
6

 

 

15
ELECTROMAGNETIC
150
69
6
7
GRAVITY
102
39
3
22
Add to Reduce
252
108
9
2+2
Reduce to Deduce
2+5+2
1+0+8
-
4
Essence of Number
9
9
9

 

 

15
ELECTROMAGNETIC
150
69
6
5
LIGHT
56
29
2
20
First Total
206
98
8
2+0
Add to Reduce
2+0+6
9+8
-
2
Second Total
8
17
8
-
Reduce to Deduce
-
1+7
-
2
Essence of Number
8
8
8

 

 

15
ELECTROMAGNETIC
150
69
6
6
LIGHTS
75
30
3
21
First Total
225
99
9
2+1
Add to Reduce
2+2+5
9+9
-
3
Second Total
9
18
9
-
Reduce to Deduce
-
1+8
-
3
Essence of Number
9
9
9

 

 

-
21
E
L
E
C
T
R
O
-
M
A
G
N
E
T
I
C
-
L
I
G
H
T
S
-
-
-
-
-
--
-
-
-
-
38
-
-
-
-
-
-
-
6
-
-
-
-
5
-
-
9
-
-
-
9
-
8
-
1
+
=
38
3+8
=
11
1+1
2
=
2
74
`-
-
-
-
-
-
-
15
-
-
-
-
14
-
-
9
-
-
-
9
-
8
-
19
+
=
74
7+4
=
11
1+1
2
=
2
-
21
E
L
E
C
T
R
O
-
M
A
G
N
E
T
I
C
-
L
I
G
H
T
S
-
-
-
-
-
--
-
-
-
-
52
-
5
3
5
3
2
9
-
-
4
1
7
-
5
2
-
3
-
3
-
7
-
2
-
+
=
52
5+2
=
7
=
7
=
7
106
`-
5
12
5
3
20
18
-
-
13
1
7
-
5
20
-
3
-
12
-
7
-
20
-
+
=
106
1+0+6
=
7
=
7
=
7
-
21
E
L
E
C
T
R
O
-
M
A
G
N
E
T
I
C
-
L
I
G
H
T
S
-
-
-
-
-
--
-
-
-
-
225
`-
5
12
5
3
20
18
15
-
13
1
7
14
5
20
9
3
-
12
9
7
8
20
19
+
=
225
2+2+5
=
9
=
9
=
9
99
-
5
3
5
3
2
9
6
-
4
1
7
5
5
2
9
3
-
3
9
7
8
2
1
+
=
99
9+9
=
18
1+8
9
=
9
-
21
E
L
E
C
T
R
O
-
M
A
G
N
E
T
I
C
-
L
I
G
H
T
S
-
-
-
-
-
--
-
-
-
-
1
-
-
1
1
-
-
-
1
-
-
1
1
-
-
-
-
-
-
1
-
-
-
-
1-
-
-
1
occurs
x
2
=
2
=
2
2
-
-
-
-
-
2
--
-
-
-
--
-
4
-
2
-
-
-
-
4
-
-
2
-
-
-
2
occurs
x
3
=
6
=
6
3
-
-
3
-
3
-
-
-
-
-
-3
-
-
-
-
-
3
-
3
-
-
-
-
-
-
-
3
occurs
x
4
=
12
=
3
4
-
-
-
-
-
-
--
-
-
4
--
-
4
-
-
-
-
-
-
4
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
5
-
5
-
5
5
-
-
-
-
-
-
-
5
5
5
-
5
-
-
-
-
5
-
5
-
-
5
occurs
x
4
=
20
=
2
6
-
-
-
-
-
-
-
6
-
-
-
-
-
3
-
2
-
-
-
-
3
-
2
-
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
-
-
-
-
-
-
-
7
-
3
-
-
-
-
-
-
7
-
-
-
-
-
7
occurs
x
2
=
14
=
5
8
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
2
-
-
-
-
3
8
2
-
-
-
8
occurs
x
1
=
8
=
8
9
-
9
-
-
-
9
9
-
-
9
-9
-
-
-
-
9
-
-
-
9
-
-
-
-
-
-
9
occurs
x
3
=
27
=
9
45
21
E
L
E
C
T
R
O
-
M
A
G
N
E
T
I
C
-
L
I
G
H
T
S
-
-
45
-
-
21
-
99
-
45
4+5
-
-
-
-
-
-
9
-
-
-
-
-
-
-
9
-
-
-
9
-
-
-
-
-
-
4+5
-
-
2+1
-
9+9
-
4+5
9
21
E
L
E
C
T
R
O
-
M
A
G
N
E
T
I
C
-
L
I
G
H
T
S
-
-
9
-
-
3
-
18
-
9
-
-
5
3
5
3
2
9
6
-
4
1
7
5
5
2
9
3
-
3
9
7
8
2
1
-
-
-
-
-
-
-
1+8
-
-
9
21
E
L
E
C
T
R
O
-
M
A
G
N
E
T
I
C
-
L
I
G
H
T
S
-
-
9
-
-
3
-
9
-
9

 

 

-
-
-
-
-
LIGHT
-
-
-
-
-
-
-
4
LIGH
36
27
9
-
-
-
-
1
T
20
2
2
L
=
3
-
5
LIGHT
56
29
11
-
-
1+0
-
-
-
5+6
2+9
1+1
-
-
1
-
9
-
11
11
7
-
-
1+0
-
-
-
1+1
1+1
-
-
-
1
-
9
-
2
2
2

 

 

-
5
L
I
G
H
T
-
-
-
-
-
--
-
-
-
-
-
`-
-
9
-
8
-
+
=
17
1+7
=
8
-
8
-
8
-
5
L
I
G
H
T
-
-
-
-
-
--
-
-
-
-
-
-
3
-
7
-
2
+
=
12
1+2
=
3
=
3
-
3
-
`-
12
-
7
-
20
+
=
39
3+9
=
12
1+2
3
-
3
-
5
L
I
G
H
T
-
-
-
-
-
--
-
-
-
-
-
`-
12
9
7
8
20
+
=
56
5+6
=
11
1+1
2
-
2
-
-
3
9
7
8
2
+
=
29
2+9
=
11
1+1
2
-
2
-
5
L
I
G
H
T
-T
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
2
occurs
x
1
=
2
=
2
-
-
3
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
4
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
--
-
-
7
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
-
-
-
8
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
9
-
-
-
-
-
9
occurs
x
1
=
9
=
9
16
5
L
I
G
H
T
-
-
29
-
1
5
-
29
-
29
1+6
-
-
9
-
-
-
-
-
2+9
-
-
-
-
2+9
-
2+9
7
5
L
I
G
H
T
-
-
11
-
1
5
-
11
-
11
-
-
3
9
7
8
2
-
-
-
-
-
-
-
-
-
-
7
5
L
I
G
H
T
-
-
2
-
-
5
-
2
-
2

 

 

5
L
I
G
H
T
-
-
-
-
-
--
-
-
-
-
`-
-
9
-
8
-
+
=
17
1+7
=
8
-
8
-
8
5
L
I
G
H
T
-
-
-
-
-
--
-
-
-
-
-
3
-
7
-
2
+
=
12
1+2
=
3
=
3
-
3
`-
12
-
7
-
20
+
=
39
3+9
=
12
1+2
3
-
3
5
L
I
G
H
T
-
-
-
-
-
--
-
-
-
-
`-
12
9
7
8
20
+
=
56
5+6
=
11
1+1
2
-
2
-
3
9
7
8
2
+
=
29
2+9
=
11
1+1
2
-
2
5
L
I
G
H
T
-T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
2
occurs
x
1
=
2
=
2
-
3
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
7
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
-
-
8
-
-
-
8
occurs
x
1
=
8
=
8
-
-
9
-
-
-
-
-
9
occurs
x
1
=
9
=
9
5
L
I
G
H
T
-
-
29
-
1
5
-
29
-
29
-
-
9
-
-
-
-
-
2+9
-
-
-
-
2+9
-
2+9
5
L
I
G
H
T
-
-
11
-
1
5
-
11
-
11
-
3
9
7
8
2
-
-
-
-
-
-
-
-
-
-
5
L
I
G
H
T
-
-
2
-
-
5
-
2
-
2

 

 

P
=
7
-
4
PARTICLES
103
40
4
O
=
6
-
1
OF
21
12
3
L
=
3
-
5
LIGHT
56
29
2
-
-
16
-
10
-
180
81
9
-
-
1+6
-
1+0
-
1+8+0
8+1
-
-
-
7
-
1
-
9
9
9

 

 

9
PARTICLES
103
40
4
2
OF
21
12
3
5
LIGHT
56
29
2
16
Add to Reduce
90
81
9
1+6
Reduce to Deduce
9+0
8+1
-
7
Essence of Number
9
9
9

 

 

-
16
P
A
R
T
I
C
L
E
S
-
O
F
-
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
1
-
6
-
-
-
9
-
8
-
+
=
33
3+3
=
6
-
6
-
6
-
-
-
-
-
-
9
-
-
-
19
-
15
-
-
-
9
-
8
-
+
=
60
6+0
=
6
-
6
-
6
-
16
P
A
R
T
I
C
L
E
S
-
O
F
-
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
-
7
1
9
2
-
3
3
5
-
-
-
6
-
3
-
7
-
2
+
=
48
4+8
=
12
1+2
3
-
3
-
`-
16
1
18
20
-
3
12
5
-
-
-
6
-
12
-
7
-
20
+
=
120
1+2+0
=
3
-
3
-
3
-
16
P
A
R
T
I
C
L
E
S
-
O
F
-
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
-
16
1
18
20
9
3
12
5
19
-
15
6
-
12
9
7
8
20
+
=
180
1+8+0
=
9
-
9
-
9
-
-T
7
1
9
2
9
3
3
5
1
-T
6
6
-T
3
9
7
8
2
+
=
81
8+1
=
9
-
9
-
9
-
16
P
A
R
T
I
C
L
E
S
-
O
F
-
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
-
3
3
-
-
-
-
-
-
3
-
-
-
-
-
-
3
occurs
x
3
=
9
=
9
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
-
5
occurs
x
1
=
5
=
5
-
-
-
-
-
-
-
-
-
-
-
-
6
6
-
-
-
-
-
-
-
-
6
occurs
x
2
=
12
1+2
3
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
7
occurs
x
2
=
14
1+4
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
9
-
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
9
occurs
x
3
=
27
=
9
4
16
P
A
R
T
I
C
L
E
S
-
O
F
-
L
I
G
H
T
-
-
41
-
1
16
-
81
-
45
-
-
-
-
9
-
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
4+1
-
-
1+6
-
8+1
-
4+5
4
16
P
A
R
T
I
C
L
E
S
-
O
F
-
L
I
G
H
T
-
-
5
-
1
7
-
9
-
9
-
-
7
1
9
2
9
3
-
5
1
-
6
6
-
3
9
7
8
2
-
-
-
-
-
-
-
-
-
-
4
16
P
A
R
T
I
C
L
E
S
-
O
F
-
L
I
G
H
T
-
-
5
-
-
7
-
9
-
9

 

 

-
0
L
I
G
H
T
-
W
A
V
E
S
-
-
-
-
-
--
-
-
-
-
-
`-
-
9
-
8
-
-
-
-
-
-
1
+
=
18
-
=
9
=
9
=
9
-
`-
-
9
-
8
-
-
-
-
-
-
19
+
=
38
-
=
9
=
9
=
9
-
10
L
I
G
H
T
-
W
A
V
E
S
-
-
-
-
-
--
-
-
-
-
-
-
3
-
7
-
2
-
5
1
4
5
-
+
=
27
2+7
=
9
=
9
=
9
-
`-
12
-
7
-
20
-
23
1
22
5
-
+
=
90
7+5
=
9
=
9
=
9
-
10
L
I
G
H
T
-
W
A
V
E
S
-T
-
-
-
-
-
-
-
-
-
-
`-
12
9
7
8
20
-
23
1
22
5
19
+
=
126
1+2+6
=
9
=
9
=
9
-T
-
3
9
7
8
2
-T
5
1
4
5
1
+
=
45
4+5
=
9
=
9
=
9
-
10
L
I
G
H
T
-
W
A
V
E
S
-T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
-
5
-
-
5
-
-
-
5
occurs
x
2
=
10
1+0
1
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
6
10
L
I
G
H
T
-
W
A
V
E
S
-
-
39
-
1
12
-
45
-
36
-
1+0
-
9
-
-
-
-
-
-
-
-
-
-
-
3+9
-
-
1+2
-
4+5
-
3+6
-
1
L
I
G
H
T
-
W
A
V
E
S
-
-
12
-
1
3
-
9
-
9
-
-
3
9
7
8
2
-
5
1
4
5
1
-
-
1+2
-
-
-
-
-
-
-
6
1
L
I
G
H
T
-
W
A
V
E
S
-
-
3
-
-
3
-
9
-
9

 

 

5
WAVES
70
16
7
9
PARTICLES
103
40
4

 

 

4
WAVE
51
15
6
9
PARTICLES
103
40
4
13
First Total
154
55
10
1+3
Add to Reduce
1+5+4
5+5
1+0
4
Second Total
10
10
1
-
Reduce to Deduce
1+0
1+0
-
4
Essence of Number
1
1
1

 

 

W
=
5
4
WAVE
51
15
6
P
=
7
8
PARTICLE
84
39
3
-
-
12
12
Add to Reduce
135
54
9
-
-
 1+2 1+2
Reduce to Deduce
1+3+5
5+4
-
-
-
3
3
Essence of Number
9
9
9

 

 

-
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
+
=
9
-
=
9
-
9
-
9
-
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
--
-
-
-
-
-
-
5
1
4
5
-
7
1
9
2
-
3
3
5
+
=
45
4+5
=
9
-
9
-
9
-
-
23
1
22
5
-
16
1
18
20
-
3
12
5
+
=
126
1+2+6
=
9
-
9
-
9
-
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
--
-
-
-
-
-
-
23
1
22
5
-
16
1
18
20
9
3
12
5
+
=
135
1+3+5
=
9
-
9
=
9
-
-
5
1
4
5
-
7
1
9
2
9
3
3
5
+
=
54
5+4
=
9
-
9
=
9
-
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
-
2
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
-
-
-
-
-
-
-
-
3
3
-
-
-
3
occurs
x
2
=
6
-
6
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
-
4
-
-
5
-
-
5
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
3
=
15
1+5
6
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
-
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
9
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
14
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
31
-
-
12
-
54
-
36
1+4
1+2
-
-
-
-
-
-
-
9
-
9
-
-
-
-
-
3+1
-
-
1+2
-
5+4
-
3+6
5
3
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
4
-
-
3
-
9
-
9
-
-
5
1
4
5
-
7
1
9
2
9
3
3
5
-
-
-
-
-
-
-
-
-
-
5
3
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
4
-
-
3
-
9
-
9

 

 

15
ELECTROMAGNETIC
150
69
6
5
LIGHT
56
29
2
20
First Total
206
98
8
2+0
Add to Reduce
2+0+6
9+8
-
2
Second Total
8
17
8
-
Reduce to Deduce
-
1+7
-
2
Essence of Number
8
8
8

 

 

15
ELECTROMAGNETIC
150
69
6
6
LIGHTS
75
30
3
21
First Total
225
99
9
2+1
Add to Reduce
2+2+5
9+9
-
3
Second Total
9
18
9
-
Reduce to Deduce
-
1+8
-
3
Essence of Number
9
9
9

 

 

-
13
M
A
G
N
E
T
I
C
-
F
I
E
L
D
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
5
-
-
9
-
-
-
9
-
-
-
+
=
23
2+3
=
5
=
5
=
5
-
-
-
-
-
14
-
-
9
-
-
-
9
-
-
-
+
=
32
3+2
=
5
=
5
=
5
-
13
M
A
G
N
E
T
I
C
-
F
I
E
L
D
-
-
-
-
-
--
-
-
-
-
-
-
4
1
7
-
5
2
-
3
-
6
-
5
3
4
+
=
40
4+0
=
4
=
4
=
4
-
-
13
1
7
-
5
20
-
3
-
6
-
5
12
4
+
=
76
7+6
=
13
1+3
4
=
4
-
13
M
A
G
N
E
T
I
C
-
F
I
E
L
D
-
-
-
-
-
--
-
-
-
-
-
-
13
1
7
14
5
20
9
3
-
6
9
5
12
4
+
=
108
1+0+8
=
9
=
9
=
9
-
-
4
1
7
5
5
2
9
3
-
6
9
5
3
4
+
=
63
6+3
=
9
=
9
=
9
-
13
M
A
G
N
E
T
I
C
-
F
I
E
L
D
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
-
-
-
-
-
3
-
-
-
-
3
-
-
-
3
occurs
x
2
=
6
=
6
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
2
=
8
=
8
-
-
-
-
-
5
5
-
-
-
-
-
-
5
-
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
-
-
-
-
-
6
--
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
9
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
8
13
M
A
G
N
E
T
I
C
-
F
I
E
L
D
-
-
37
-
-
13
-
63
-
45
-
1+3
-
-
-
-
-
-
9
-
-
-
9
--
-
-
-
-
3+7
-
-
1+3
-
6+3
-
4+5
8
4
M
A
G
N
E
T
I
C
-
F
I
E
L
D
-
-
10
-
-
4
-
9
-
9
-
-
4
1
7
5
5
2
9
3
-
6
9
5
3
4
-
-
1+0
-
-
-
-
-
-
-
8
4
M
A
G
N
E
T
I
C
-
F
I
E
L
D
-
-
1
-
-
4
-
9
-
9

 

 

15
ELECTROMAGNETIC
150
69
6
7
GRAVITY
102
39
3
22
Add to Reduce
252
108
9
2+2
Reduce to Deduce
2+5+2
1+0+8
-
4
Essence of Number
9
9
9

 

 

8
MAGNETIC
72
36
9
4
POLE
48
21
3

 

 

5
NORTH
75
30
3
5
SOUTH
83
20
2

 

 

8
MAGNETIC
72
36
9
5
POLES
67
22
4

 

 

M
=
4
=
8
MAGNETIC
72
36
9
F
=
6
=
8
FLUX
63
39
9
-
-
10
-
16
Add to Reduce
135
54
9
-
-
1+0
-
1+6
Reduce to Deduce
1+3+5
5+4
-
-
-
1
-
7
Essence of Number
9
9
9

 

 

-
12
M
A
G
N
E
T
I
C
-
F
L
U
X
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
5
-
-
9
-
-
-
-
-
6
+
=
20
2+0
=
2
=
2
=
2
-
-
-
-
-
14
-
-
9
-
-
-
-
-
24
+
=
47
4+7
=
11
1+1
2
=
2
-
12
M
A
G
N
E
T
I
C
-
F
L
U
X
-
-
-
-
-
--
-
-
-
-
-
-
4
1
7
-
5
2
-
3
-
6
3
3
-
+
=
34
3+4
=
7
=
7
=
7
-
-
13
1
7
-
5
20
-
3
-
6
12
21
-
+
=
88
8+8
=
16
1+6
7
=
7
-
12
M
A
G
N
E
T
I
C
-
F
L
U
X
-
-
-
-
-
--
-
-
-
-
-
-
13
1
7
14
5
20
9
3
-
6
12
21
24
+
=
135
1+3+5
=
9
=
9
=
9
-
-
4
1
7
5
5
2
9
3
-
6
3
3
6
+
=
54
5+4
=
9
=
9
=
9
-
12
M
A
G
N
E
T
I
C
-
F
L
U
X
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
-
-
-
-
-
3
-
-
3
3
-
-
-
3
occurs
x
3
=
9
=
9
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
5
5
-
-
-
-
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
-
-
-
-
-
-
6
-
-
6
-
-
6
occurs
x
2
=
12
1+2
3
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
8
12
M
A
G
N
E
T
I
C
-
F
L
U
X
-
-
37
-
-
12
-
54
-
36
-
1+2
-
-
-
5
5
-
-
-
-
-
-
-
-
-
-
3+7
-
-
1+2
-
5+4
-
3+6
8
3
M
A
G
N
E
T
I
C
-
F
L
U
X
-
-
10
-
-
3
-
9
-
9
-
-
4
1
7
5
5
2
9
3
-
6
3
3
6
-
-
1+0
-
-
-
-
-
-
-
8
3
M
A
G
N
E
T
I
C
-
F
L
U
X
-
-
1
-
-
3
-
9
-
9

 

 

-
4
F
L
U
X
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
6
+
=
5
-
=
6
=
6
=
6
-
-
-
-
-
24
+
=
24
2+4
=
6
=
6
=
6
-
4
F
L
U
X
-
-
-
-
-
--
-
-
-
-
-
-
6
3
3
-
+
=
12
1+2
=
3
=
3
=
3
-
-
6
12
21
-
+
=
39
3+9
=
12
1+2
3
=
3
-
4
F
L
U
X
-
-
-
-
-
--
-
-
-
-
-
-
6
12
21
24
+
=
72
7+2
=
9
=
9
=
9
-
-
6
3
3
6
+
=
18
1+8
=
9
=
9
=
9
-
4
F
L
U
X
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
ONE
1
-
-
-
-
-
2
-
-
-
-
-
-
-
2
TWO
2
-
-
-
-
-
-
-
-
3
3
-
-
-
3
occurs
x
2
=
6
=
6
4
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
5
-
-
-
-
-
-
-
5
FIVE
5
-
-
-
-
-
-
-
6
-
-
6
-
-
6
occurs
x
2
=
12
1+2
3
7
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
8
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
9
-
-
-
-
-
-
-
9
NINE
9
-
-
-
-
-
36
4
F
L
U
X
-
-
9
-
-
4
-
18
-
9
3+6
-
-
-
-
-
-
-
`-
-
-
-
-
1+8
-
`-
9
4
F
L
U
X
-
-
9
-
-
4
-
9
-
9
-
-
6
3
3
6
-
-
-
-
-
-
-
-
-
-
9
4
F
L
U
X
-
-
9
-
-
4
-
9
-
9

 

 

4
F
L
U
X
-
-
-
-
-
--
-
-
-
-
-
-
-
-
6
+
=
5
-
=
6
=
6
=
6
-
-
-
-
24
+
=
24
2+4
=
6
=
6
=
6
4
F
L
U
X
-
-
-
-
-
--
-
-
-
-
-
6
3
3
-
+
=
12
1+2
=
3
=
3
=
3
-
6
12
21
-
+
=
39
3+9
=
12
1+2
3
=
3
4
F
L
U
X
-
-
-
-
-
--
-
-
-
-
-
6
12
21
24
+
=
72
7+2
=
9
=
9
=
9
-
6
3
3
6
+
=
18
1+8
=
9
=
9
=
9
4
F
L
U
X
-
-
-
-
-
-
-
-
-
-
-
-
3
3
-
-
-
3
occurs
x
2
=
6
=
6
-
6
-
-
6
-
-
6
occurs
x
2
=
12
1+2
3
4
F
L
U
X
-
-
9
-
-
4
-
18
-
9
-
-
-
-
-
-
-
`-
-
-
-
-
1+8
-
`-
4
F
L
U
X
-
-
9
-
-
4
-
9
-
9
-
6
3
3
6
-
-
-
-
-
-
-
-
-
-
4
F
L
U
X
-
-
9
-
-
4
-
9
-
9

 

 

-
VACUUM
-
-
-
2
V+A
23
5
5
3
C+U+U
45
9
9
1
M
13
4
4
6
VACUUM
81
18
18
-
-
8+1
1+8
1+8
6
VACUUM
9
9
9

 

 

-
6
V
A
C
U
U
M
-
-
-
-
-
--
-
-
-
-
4
1
3
3
3
4
+
=
18
1+8
=
9
=
9
-
-
22
1
3
21
21
13
+
=
81
8+1
=
9
=
9
-
6
V
A
C
U
U
M
-
-
-
-
-
--
-
-
-
-
22
1
3
21
21
13
+
=
81
8+1
=
9
=
9
-
-
4
1
3
3
3
4
+
=
18
1+8
=
9
=
9
-
6
V
A
C
U
U
M
-
-
-
-
-
--
-
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
1
=
1
2
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
3
3
3
-
-
-
3
occurs
x
3
=
9
-
-
4
-
-
-
-
4
-
-
4
occurs
x
2
=
8
5
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
37
6
V
A
C
U
U
M
-
-
8
-
-
6
-
18
3+7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1+8
10
6
V
A
C
U
U
M
-
-
8
-
-
6
-
9
1+0
4
1
3
3
3
4
4
-
-
-
-
-
--
-
-
1
6
V
A
C
U
U
M
-
-
8
-
-
6
-
9

 

 

6
V
A
C
U
U
M
-
-
-
-
-
--
-
-
-
4
1
3
3
3
4
+
=
18
1+8
=
9
=
9
-
22
1
3
21
21
13
+
=
81
8+1
=
9
=
9
6
V
A
C
U
U
M
-
-
-
-
-
--
-
-
-
22
1
3
21
21
13
+
=
81
8+1
=
9
=
9
-
4
1
3
3
3
4
+
=
18
1+8
=
9
=
9
6
V
A
C
U
U
M
-
-
-
-
-
--
-
-
-
-
1
-
-
-
-
-
-
1
occurs
x
1
=
1
-
-
-
3
3
3
-
-
-
3
occurs
x
3
=
9
-
4
-
-
-
-
4
-
-
4
occurs
x
2
=
8
-
-
-
-
-
-
-
-
9
-
-
-
-
-
6
V
A
C
U
U
M
-
-
8
-
-
6
-
18
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1+8
6
V
A
C
U
U
M
-
-
8
-
-
6
-
9
4
1
3
3
3
4
4
-
-
-
-
-
--
-
-
6
V
A
C
U
U
M
-
-
8
-
-
6
-
9

 

 

-
11
N
O
T
H
I
N
G
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
-
5
6
-
8
9
5
-
5
-
1
1
+
=
40
4+0
=
4
=
4
=
4
-
-``
14
15
-
8
9
14
-
14
-
19
19
+
=
112
1+1+2
=
4
=
4
=
4
-
11
N
O
T
H
I
N
G
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
-
-
-
2
-
-
-
7
-
5
-
-
+
=
14
1+4
=
5
=
5
=
5
-
-
-
-
20
-
-
-
7
-
5
-
-
+
=
32
3+2
=
5
=
5
=
5
-
11
N
O
T
H
I
N
G
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
-
14
15
20
8
9
14
7
14
5
19
19
+
=
144
1+4+4
=
9
=
9
=
9
-
-
5
6
2
8
9
5
7
5
5
1
1
+
=
54
5+4
=
9
=
9
=
9
-
11
N
O
T
H
I
N
G
N
E
S
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
1
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
5
-
-
-
-
5
-
5
5
-
-
-
-
5
occurs
x
4
=
20
2+0
2
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
-
-
9
-
-
-
=-
-
-
-
-
9
occurs
x
1
=
9
=
9
7
11
N
O
T
H
I
N
G
N
E
S
S
-
-
38
-
-
11
-
54
-
36
-
1+1
--
-
-
-
9
-
-
-
-
-
-
-
-
3+8
-
-
1+1
-
5+4
-
3+6
7
2
N
O
T
H
I
N
G
N
E
S
S
-
-
11
-
-
2
-
9
-
9
-
-
5
6
2
8
9
5
7
5
5
1
1
-
-
1+1
-
-
=
-
-
-
-
7
2
N
O
T
H
I
N
G
N
E
S
S
-
-
2
-
-
2
-
9
-
9

 

 

11
N
O
T
H
I
N
G
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
5
6
-
8
9
5
-
5
-
1
1
+
=
40
4+0
=
4
=
4
=
4
-``
14
15
-
8
9
14
-
14
-
19
19
+
=
112
1+1+2
=
4
=
4
=
4
11
N
O
T
H
I
N
G
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
-
-
2
-
-
-
7
-
5
-
-
+
=
14
1+4
=
5
=
5
=
5
-
-
-
20
-
-
-
7
-
5
-
-
+
=
32
3+2
=
5
=
5
=
5
11
N
O
T
H
I
N
G
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
14
15
20
8
9
14
7
14
5
19
19
+
=
144
1+4+4
=
9
=
9
=
9
-
5
6
2
8
9
5
7
5
5
1
1
+
=
54
5+4
=
9
=
9
=
9
11
N
O
T
H
I
N
G
N
E
S
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
1
-
-
1
occurs
x
2
=
2
=
2
-
-
-
2
-
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
5
-
-
-
-
5
-
5
5
-
-
-
-
5
occurs
x
4
=
20
2+0
2
-
-
6
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
-
-
-
-
-
7
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
-
-
8
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
-
9
-
-
-
=-
-
-
-
-
9
occurs
x
1
=
9
=
9
11
N
O
T
H
I
N
G
N
E
S
S
-
-
38
-
-
11
-
54
-
36
1+1
5
-
-
-
-
5
-
5
5
-
-
-
-
3+8
-
-
1+1
-
5+4
-
3+6
2
N
O
T
H
I
N
G
N
E
S
S
-
-
11
-
-
2
-
9
-
9
-
5
6
2
8
9
5
7
5
5
1
1
-
-
1+1
-
-
=
-
-
-
-
2
N
O
T
H
I
N
G
N
E
S
S
-
-
2
-
-
2
-
9
-
9

 

 

-
8
F
U
L
L
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
5
-
1
1
+
=
7
-
=
7
=
7
=
7
-
-``
-
-
-
-
14
-
19
19
+
=
52
5+2
=
7
=
7
=
7
-
8
F
U
L
L
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
-
6
3
3
3
-
5
-
-
+
=
20
2+0
=
2
-
2
=
2
-
-
6
21
12
12
-
5
-
-
+
=
56
5+6
=
11
1+1
2
=
2
-
8
F
U
L
L
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
-
6
21
12
12
14
5
19
19
+
=
108
1+0+8
=
9
=
9
=
9
-
-
6
3
3
3
5
5
1
1
+
=
27
2+7
=
9
=
9
=
9
-
8
F
U
L
L
N
E
S
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
1
-
-
1
occurs
x
2
=
2
=
2
2
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
3
3
3
-
-
-
-
-
-
3
occurs
x
3
=
9
=
9
4
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
5
5
-
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
6
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
30
8
F
U
L
L
N
E
S
S
-
-
15
-
-
8
-
27
-
18
3+0
-
-
-
-
-
-
-
-
-
-
-
1+5
-
-
-
-
2+7
-
1+8
3
8
F
U
L
L
N
E
S
S
-
-
6
-
-
8
-
9
-
9
-
-
6
2
2
2
5
5
1
1
-
-
-
-
-
=
-
-
-
-
3
8
F
U
L
L
N
E
S
S
-
-
6
-
-
8
-
9
-
9

 

 
F
U
L
L
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
5
-
1
1
+
=
7
-
=
7
=
7
=
7
-``
-
-
-
-
14
-
19
19
+
=
52
5+2
=
7
=
7
=
7
8
F
U
L
L
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
6
3
3
3
-
5
-
-
+
=
20
2+0
=
2
-
2
=
2
-
6
21
12
12
-
5
-
-
+
=
56
5+6
=
11
1+1
2
=
2
8
F
U
L
L
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
6
21
12
12
14
5
19
19
+
=
108
1+0+8
=
9
=
9
=
9
-
6
3
3
3
5
5
1
1
+
=
27
2+7
=
9
=
9
=
9
8
F
U
L
L
N
E
S
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
1
-
-
1
occurs
x
2
=
2
=
2
-
-
3
3
3
-
-
-
-
-
-
3
occurs
x
3
=
9
=
9
-
-
-
-
-
5
5
-
-
-
-
5
occurs
x
2
=
10
1+0
1
-
6
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
=
6
8
F
U
L
L
N
E
S
S
-
-
15
-
-
8
-
27
-
18
-
-
-
-
-
-
-
-
-
-
-
1+5
-
-
-
-
2+7
-
1+8
8
F
U
L
L
N
E
S
S
-
-
6
-
-
8
-
9
-
9
-
6
2
2
2
5
5
1
1
-
-
-
-
-
=
-
-
-
-
8
F
U
L
L
N
E
S
S
-
-
6
-
-
8
-
9
-
9

 

 

-
19
N
O
T
H
I
N
G
N
E
S
S
-
F
U
L
L
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
-
5
6
-
8
9
5
-
5
-
1
1
-
-
-
-
-
5
-
1
1
+
=
47
4+7
=
11
1+1
2
=
2
-
-``
14
15
-
8
9
14
-
14
-
19
19
-
-
-
-
-
14
-
19
19
+
=
164
1+6+4
=
11
1+1
2
=
2
-
19
N
O
T
H
I
N
G
N
E
S
S
-
F
U
L
L
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
-
-
-
2
-
-
-
7
-
5
-
-
-
6
3
3
3
-
5
-
-
+
=
34
3+4
=
7
-
7
=
7
-
-
-
-
20
-
-
-
7
-
5
-
-
-
6
21
12
12
-
5
-
-
+
=
88
8+8
=
16
1+6
7
=
7
-
19
N
O
T
H
I
N
G
N
E
S
S
-
F
U
L
L
N
E
S
S
-
-
-
-
-
--
-
-
-
-
-
-
14
15
20
8
9
14
7
14
5
19
19
-
6
21
12
12
14
5
19
19
+
=
252
2+5+2
=
9
=
9
=
9
-
-
5
6
2
8
9
5
7
5
5
1
1
-
6
3
3
3
5
5
1
1
+
=
81
8+1
=
9
=
9
=
9
-
19
N
O
T
H
I
N
G
N
E
S
S
-
F
U
L
L
N
E
S
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
1
-
-
-
-
-
-
-
1
1
-
-
1
occurs
x
4
=
4
=
4
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
-
-
-
-
-
-
-
-
-
--
-
3
3
3
-
-
-
-
-
-
3
occurs
x
3
=
9
=
9
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
5
-
-
-
-
5
-
5
5
-
-
-
-
-
-
-
5
5
-
-
-
-
5
occurs
x
6
=
30
3+0
3
-
-
-
6
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
6
occurs
x
2
=
12
1+2
3
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
-
-
9
-
-
-
=-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
4
19
N
O
T
H
I
N
G
N
E
S
S
-
F
U
L
L
N
E
S
S
-
-
41
-
-
19
-
81
-
45
-
1+9
5
-
-
-
-
5
-
5
5
-
-
-
-
-
-
-
5
5
-
-
-
-
4+1
-
-
1+9
-
8+1
-
4+5
4
10
N
O
T
H
I
N
G
N
E
S
S
-
F
U
L
L
N
E
S
S
-
-
5
-
-
10
-
9
-
9
-
1+0
5
6
2
8
9
5
7
5
5
1
1
-
6
2
2
2
5
5
1
1
-
-
-
-
-
1+0
-
-
-
-
4
1
N
O
T
H
I
N
G
N
E
S
S
-
F
U
L
L
N
E
S
S
-
-
5
-
-
1
-
9
-
9

 

 

-
6
E
N
E
R
G
Y
-
-
-
-
-
--
-
-
-
-
-
-
-
5
-
-
-
-
+
=
5
-
=
5
=
5
-
5
-
-
-
14
-
-
-
-
+
=
14
1+4
=
5
=
5
-
5
-
6
E
N
E
R
G
Y
-
-
-
-
-
-
-
-
-
-
-
-
5
-
5
9
7
7
+
=
33
3+3
=
6
=
6
-
6
-
-
5
-
5
18
7
25
+
=
60
6+0
=
6
=
6
-
6
-
6
E
N
E
R
G
Y
-
-
-
-
-
-
-
-
-
-
-
-
5
14
5
18
7
25
+
=
74
7+4
=
11
1+1
2
-
2
-
-
5
5
5
9
7
7
+
=
38
3+8
=
11
1+1
2
-
2
-
6
E
N
E
R
G
Y
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
5
5
5
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
6
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
7
7
-
-
7
occurs
x
2
=
14
1+4
5
8
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
9
occurs
x
1
=
9
=
9
24
6
E
N
E
R
G
Y
-
-
21
-
-
6
-
38
-
20
2+4
-
5
5
5
-
-
-
-
-
2+1
-
-
-
-
3+8
-
2+0
6
6
E
N
E
R
G
Y
-
-
3
-
-
6
-
11
-
2
-
-
5
5
5
9
7
7
-
-
-
-
-
-
-
1+1
-
-
6
6
E
N
E
R
G
Y
-
-
3
-
-
6
-
2
-
2

 

 

6
E
N
E
R
G
Y
-
-
-
-
-
--
-
-
-
-
-
-
5
-
-
-
-
+
=
5
-
=
5
=
5
-
5
-
-
14
-
-
-
-
+
=
14
1+4
=
5
=
5
-
5
6
E
N
E
R
G
Y
-
-
-
-
-
-
-
-
-
-
-
5
-
5
9
7
7
+
=
33
3+3
=
6
=
6
-
6
-
5
-
5
18
7
25
+
=
60
6+0
=
6
=
6
-
6
6
E
N
E
R
G
Y
-
-
-
-
-
-
-
-
-
-
-
5
14
5
18
7
25
+
=
74
7+4
=
11
1+1
2
-
2
-
5
5
5
9
7
7
+
=
38
3+8
=
11
1+1
2
-
2
6
E
N
E
R
G
Y
-
-
-
-
-
-
-
-
-
-
-
5
5
5
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
7
7
-
-
7
occurs
x
2
=
14
1+4
5
-
-
-
-
9
-
-
-
-
9
occurs
x
1
=
9
=
9
6
E
N
E
R
G
Y
-
-
21
-
-
6
-
38
-
20
-
5
5
5
-
-
-
-
-
2+1
-
-
-
-
3+8
-
2+0
6
E
N
E
R
G
Y
-
-
3
-
-
6
-
11
-
2
-
5
5
5
9
7
7
-
-
-
-
-
-
-
1+1
-
-
6
E
N
E
R
G
Y
-
-
3
-
-
6
-
2
-
2

 

 

-
12
A
T
O
M
I
C
-
E
N
E
R
G
Y
-
-
-
-
--
-
-
-
-
-
-
-
-
6
-
9
-
-
-
5
-
-
-
-
+
=
20
2+0
=
2
=
2
=
2
-
-
-
-
15
-
9
-
-
-
14
-
-
-
-
+
=
38
3+8
=
11
1+1
2
=
2
-
12
A
T
O
M
I
C
-
E
N
E
R
G
Y
-
-
-
-
-
--
-
-
-
-
-
-
1
2
-
4
-
3
-
5
-
5
9
7
7
+
=
43
4+3
=
5
=
7
=
7
-
-
1
20
-
13
-
3
-
5
-
5
18
7
25
+
=
97
9+7
=
16
1+6
7
=
7
-
12
A
T
O
M
I
C
-
E
N
E
R
G
Y
-
-
-
-
-
--
-
-
-
-
-
-
1
20
15
13
9
3
-
5
14
5
18
7
25
+
=
135
1+3+5
=
9
=
9
=
9
-
-
1
2
6
4
9
3
-
5
5
5
9
7
7
+
=
63
6+3
=
9
=
9
=
9
-
12
A
T
O
M
I
C
-
E
N
E
R
G
Y
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
--
-
-
-
3
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
-
-
5
5
5
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
1+2
6
-
--
--
--
--
--
--
--
--
-
-
-
-
7
7
-
-
7
occurs
x
2
=
14
1+4
5
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
9
-
-
-
-
9
occurs
x
2
=
18
1+8
9
8
12
A
T
O
M
I
C
-
E
N
E
R
G
Y
-
-
37
-
-
12
-
63
-
36
-
1+2
-
-
-
-
-
-
-
5
5
5
-
-
-
-
-
3+7
-
-
1+2
-
6+3
-
3+6
8
3
A
T
O
M
I
C
-
E
N
E
R
G
Y
-
-
10
-
-
3
-
9
-
9
-
--
1
2
6
4
9
3
-
5
5
5
9
7
7
-
-
1+0
-
-
--
-
-
-
-
8
3
A
T
O
M
I
C
-
E
N
E
R
G
Y
-
-
1
-
-
3
-
9
-
9

 

 

12
A
T
O
M
I
C
-
E
N
E
R
G
Y
-
-
-
-
--
-
-
-
-
-
-
-
6
-
9
-
-
-
5
-
-
-
-
+
=
20
2+0
=
2
=
2
=
2
-
-
-
15
-
9
-
-
-
14
-
-
-
-
+
=
38
3+8
=
11
1+1
2
=
2
12
A
T
O
M
I
C
-
E
N
E
R
G
Y
-
-
-
-
-
--
-
-
-
-
-
1
2
-
4
-
3
-
5
-
5
9
7
7
+
=
43
4+3
=
5
=
7
=
7
-
1
20
-
13
-
3
-
5
-
5
18
7
25
+
=
97
9+7
=
16
1+6
7
=
7
12
A
T
O
M
I
C
-
E
N
E
R
G
Y
-
-
-
-
-
--
-
-
-
-
-
1
20
15
13
9
3
-
5
14
5
18
7
25
+
=
135
1+3+5
=
9
=
9
=
9
-
1
2
6
4
9
3
-
5
5
5
9
7
7
+
=
63
6+3
=
9
=
9
=
9
12
A
T
O
M
I
C
-
E
N
E
R
G
Y
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
--
-
-
-
3
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
-
5
5
5
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
1+2
6
--
--
--
--
--
--
--
--
-
-
-
-
7
7
-
-
7
occurs
x
2
=
14
1+4
5
-
-
-
-
-
9
-
-
-
-
-
9
-
-
-
-
9
occurs
x
2
=
18
1+8
9
12
A
T
O
M
I
C
-
E
N
E
R
G
Y
-
-
37
-
-
12
-
63
-
36
1+2
-
-
-
-
-
-
-
5
5
5
-
-
-
-
-
3+7
-
-
1+2
-
6+3
-
3+6
3
A
T
O
M
I
C
-
E
N
E
R
G
Y
-
-
10
-
-
3
-
9
-
9
--
1
2
6
4
9
3
-
5
5
5
9
7
7
-
-
1+0
-
-
--
-
-
-
-
3
A
T
O
M
I
C
-
E
N
E
R
G
Y
-
-
1
-
-
3
-
9
-
9

 

 

-
12
A
T
O
M
I
C
E
N
E
R
G
Y
-
-
-
-
--
-
-
-
-
-
-
-
-
6
-
9
-
-
5
-
-
-
-
+
=
20
2+0
=
2
=
2
=
2
-
-
-
-
15
-
9
-
-
14
-
-
-
-
+
=
38
3+8
=
11
1+1
2
=
2
-
12
A
T
O
M
I
C
E
N
E
R
G
Y
-
-
-
-
-
--
-
-
-
-
-
-
1
2
-
4
-
3
5
-
5
9
7
7
+
=
43
4+3
=
5
=
7
=
7
-
-
1
20
-
13
-
3
5
-
5
18
7
25
+
=
97
9+7
=
16
1+6
7
=
7
-
12
A
T
O
M
I
C
E
N
E
R
G
Y
-
-
-
-
-
--
-
-
-
-
-
-
1
20
15
13
9
3
5
14
5
18
7
25
+
=
135
1+3+5
=
9
=
9
=
9
-
-
1
2
6
4
9
3
5
5
5
9
7
7
+
=
63
6+3
=
9
=
9
=
9
-
12
A
T
O
M
I
C
E
N
E
R
G
Y
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
--
-
-
-
3
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
-
5
5
5
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
1+2
6
-
--
--
--
--
--
--
--
-
-
-
-
7
7
-
-
7
occurs
x
2
=
14
1+4
5
-
-
-
-
-
-
9
-
-
-
-
9
-
-
-
-
9
occurs
x
2
=
18
1+8
9
8
12
A
T
O
M
I
C
E
N
E
R
G
Y
-
-
37
-
-
12
-
63
-
36
-
1+2
-
-
-
-
-
-
5
5
5
-
-
-
-
-
3+7
-
-
1+2
-
6+3
-
3+6
8
3
A
T
O
M
I
C
E
N
E
R
G
Y
-
-
10
-
-
3
-
9
-
9
-
--
1
2
6
4
9
3
5
5
5
9
7
7
-
-
1+0
-
-
--
-
-
-
-
8
3
A
T
O
M
I
C
E
N
E
R
G
Y
-
-
1
-
-
3
-
9
-
9

 

 

S
=
1
-
9
SPLITTING
126
45
9
T
=
2
-
3
THE
33
15
6
A
=
1
-
4
ATOM
49
13
4
B
-
4
4
16
First Total
208
73
19
-
-
-
-
1+6
Add to Reduce
2+0+8
7+3
1+9
-
-
7
-
7
Second Total
10
10
10
8
-
-
-
-
Reduce to Deduce
1+0
1+0
1+0
-
-
7
-
7
Essence of Number
1
1
1

 

 

-
16
S
P
L
I
T
T
I
N
G
-
T
H
E
-
A
T
O
M
-
-
-
-
-
--
-
-
-
-
38
-
1
-
-
9
-
-
9
5
-
-
-
8
-
-
-
-
6
-
+
=
38
3+8
=
11
1+1
2
=
2
83
-
19
-
-
9
-
-
9
14
-
-
-
8
-
-
-
-
15
-
+
=
74
8+3
=
11
1+1
2
=
2
-
16
S
P
L
I
T
T
I
N
G
-
T
H
E
-
A
T
O
M
-
-
-
-
-
--
-
-
-
-
25
-
-
7
3
-
2
2
-
-
7
-
2
-
5
-
1
2
-
4
+
=
35
3+5
=
8
=
8
=
8
79
-
-
16
12
-
20
20
-
-
7
-
20
-
5
-
1
20
-
13
+
=
134
1+3+4
=
8
=
8
=
8
-
16
S
P
L
I
T
T
I
N
G
-
T
H
E
-
A
T
O
M
-
-
-
-
-
--
-
-
-
-
208
-
19
16
12
9
20
20
9
14
7
-
20
8
5
-
1
20
15
13
+
=
208
2+0+8
=
10
1+0
1
=
1
73
-
1
7
3
9
2
2
9
5
7
-
2
8
5
-
1
2
6
4
+
=
73
7+3
=
10
1+0
1
=
1
-
16
S
P
L
I
T
T
I
N
G
-
T
H
E
-
A
T
O
M
-
-
-
-
-
-
-
-
-
-
1
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
2
=
2
=
2
2
``-
-
-
-
-
2
2
-
-
-
-
2
-
-
-
-
2
-
-
-
-
2
occurs
x
4
=
8
=
8
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
=
4
5
-
-
-
-
-
-
-
-
5
-
-
-
-
5
-
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
6
occurs
x
1
=
6
-
6
7
-
-
7
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
2
=
14
1+4
5
8
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
8
9
-
-
-
-
9
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
45
16
S
P
L
I
T
T
I
N
G
-
T
H
E
-
A
T
O
M
-
-
45
-
-
16
-
73
-
46
4+5
1+6
-
-
-
-
2
2
-
-
-
-
2
-
-
-
-
2
-
-
-
-
4+5
-
-
1+6
-
7+3
-
4+6
9
7
S
P
L
I
T
T
I
N
G
-
T
H
E
-
A
T
O
M
-
-
9
-
-
7
-
10
-
10
-
-
1
7
3
9
2
2
9
5
7
-
2
8
5
-
1
2
6
3
-
-
-
-
-
-
-
1+0
-
1+0
6
7
S
P
L
I
T
T
I
N
G
-
T
H
E
-
A
T
O
M
-
-
6
-
-
7
-
1
-
1

 

 

NUCLEAR UNCLEAR

 

 

-
16
S
P
L
I
T
T
I
N
G
T
H
E
A
T
O
M
-
-
-
-
-
--
-
-
-
-
38
-
1
-
-
9
-
-
9
5
-
-
8
-
-
-
6
-
+
=
38
3+8
=
11
1+1
2
=
2
83
-
19
-
-
9
-
-
9
14
-
-
8
-
-
-
15
-
+
=
74
8+3
=
11
1+1
2
=
2
-
16
S
P
L
I
T
T
I
N
G
T
H
E
A
T
O
M
-
-
-
-
-
--
-
-
-
-
25
-
-
7
3
-
2
2
-
-
7
2
-
5
1
2
-
4
+
=
35
3+5
=
8
=
8
=
8
79
-
-
16
12
-
20
20
-
-
7
20
-
5
1
20
-
13
+
=
134
1+3+4
=
8
=
8
=
8
-
16
S
P
L
I
T
T
I
N
G
T
H
E
A
T
O
M
-
-
-
-
-
--
-
-
-
-
208
-
19
16
12
9
20
20
9
14
7
20
8
5
1
20
15
13
+
=
208
2+0+8
=
10
1+0
1
=
1
73
-
1
7
3
9
2
2
9
5
7
2
8
5
1
2
6
4
+
=
73
7+3
=
10
1+0
1
=
1
-
16
S
P
L
I
T
T
I
N
G
T
H
E
A
T
O
M
-
-
-
-
-
-
-
-
-
-
1
-
1
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
2
=
2
=
2
2
``-
-
-
-
-
2
2
-
-
-
2
-
-
-
2
-
-
-
-
2
occurs
x
4
=
8
=
8
3
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
=
4
5
-
-
-
-
-
-
-
-
5
-
-
-
5
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
6
occurs
x
1
=
6
-
6
7
-
-
7
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
7
occurs
x
2
=
14
1+4
5
8
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
8
9
-
-
-
-
9
-
-
9
-
-
-
-
-
-
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
45
16
S
P
L
I
T
T
I
N
G
T
H
E
A
T
O
M
-
-
45
-
-
16
-
73
-
46
4+5
1+6
-
-
-
-
2
2
-
-
-
2
-
-
-
2
-
-
-
-
4+5
-
-
1+6
-
7+3
-
4+6
9
7
S
P
L
I
T
T
I
N
G
T
H
E
A
T
O
M
-
-
9
-
-
7
-
10
-
10
-
-
1
7
3
9
2
2
9
5
7
2
8
5
1
2
6
3
-
-
-
-
-
-
-
1+0
-
1+0
6
7
S
P
L
I
T
T
I
N
G
T
H
E
A
T
O
M
-
-
6
-
-
7
-
1
-
1

 

 

-
4
A
T
O
M
-
-
-
-
-
--
-
-
-
-
-
-
6
-
+
=
6
-
=
6
=
6
-
-
-
-
15
-
+
=
15
1+5
=
6
=
6
-
4
A
T
O
M
-
-
-
-
-
--
-
-
-
-
1
2
-
4
+
=
7
-
=
7
=
7
-
-
1
20
-
13
+
=
34
3+4
=
7
=
7
-
4
A
T
O
M
-
-
-
-
-
--
-
-
-
-
1
20
15
13
+
=
49
4+9
=
13
1+3
4
-
-
1
2
6
4
+
=
13
1+3
=
4
=
4
-
4
A
T
O
M
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
1
=
1
-
-
-
2
-
-
-
-
2
occurs
x
1
=
2
3
-
-
-
-
-
-
-
3
THREE
3
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
5
-
-
-
-
-
-
-
5
FIVE
5
-
-
-
-
-
-
-
6
-
-
-
6
occurs
x
1
=
6
7
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
8
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
9
-
-
-
-
-
-
-
9
NINE
9
-
-
-
32
4
A
T
O
M
-
-
13
-
-
4
-
13
3+2
-
-
-
-
-
-
-
1+3
-
-
-
-
1+3
5
4
A
T
O
M
-
-
4
-
-
4
-
4
-
-
1
2
6
4
-
-
-
-
-
-
-
-
5
4
A
T
O
M
-
-
4
-
-
4
-
4

 

 

4
A
T
O
M
-
-
-
-
-
--
-
-
-
-
-
6
-
+
=
6
-
=
6
=
6
-
-
-
15
-
+
=
15
1+5
=
6
=
6
4
A
T
O
M
-
-
-
-
-
--
-
-
-
1
2
-
4
+
=
7
-
=
7
=
7
-
1
20
-
13
+
=
34
3+4
=
7
=
7
4
A
T
O
M
-
-
-
-
-
--
-
-
-
1
20
15
13
+
=
49
4+9
=
13
1+3
4
-
1
2
6
4
+
=
13
1+3
=
4
=
4
4
A
T
O
M
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
1
=
1
-
-
2
-
-
-
-
2
occurs
x
1
=
2
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
-
-
6
-
-
-
6
occurs
x
1
=
6
4
A
T
O
M
-
-
13
-
-
4
-
13
-
-
-
-
-
-
-
1+3
-
-
-
-
1+3
4
A
T
O
M
-
-
4
-
-
4
-
4
-
1
2
6
4
-
-
-
-
-
-
-
-
4
A
T
O
M
-
-
4
-
-
4
-
4

 

 

-
5
A
T
O
M
S
-
-
-
-
-
--
-
-
-
-
-
-
6
-
1
+
=
7
-
=
7
=
7
-
-
-
-
15
-
19
+
=
34
3+4
=
7
=
7
-
5
A
T
O
M
S
-
-
-
-
-
--
-
-
-
-
1
2
-
4
-
+
=
7
-
=
7
=
7
-
-
1
20
-
13
-
+
=
34
3+4
=
7
=
7
-
5
A
T
O
M
S
-
-
-
-
-
--
-
-
-
-
1
20
15
13
19
+
=
68
6+8
=
14
1+4
5
-
-
1
2
6
4
1
+
=
14
1+4
=
5
=
5
-
5
A
T
O
M
S
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
-
-
1
occurs
x
2
=
2
-
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
3
-
-
-
-
-
-
-
-
3
THREE
3
-
-
-
-
-
-
-
-
4
-
-
-
4
occurs
x
1
=
4
5
-
-
-
-
-
-
-
-
5
FIVE
5
-
-
-
-
-
-
-
6
-
-
-
-
6
occurs
x
1
=
6
7
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
8
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
9
-
-
-
-
-
-
-
-
9
NINE
9
-
-
-
32
5
A
T
O
M
S
-
-
13
-
-
5
-
14
3+2
-
-
-
-
-
-
-
-
1+3
-
-
-
-
1+4
5
5
A
T
O
M
S
-
-
4
-
-
5
-
5
-
-
1
2
6
4
1
-
-
-
-
-
-
-
-
5
5
A
T
O
M
S
-
-
4
-
-
5
-
5

 

 
A
T
O
M
S
-
-
-
-
-
--
-
-
-
-
-
6
-
1
+
=
7
-
=
7
=
7
-
-
-
15
-
19
+
=
34
3+4
=
7
=
7
5
A
T
O
M
S
-
-
-
-
-
--
-
-
-
1
2
-
4
-
+
=
7
-
=
7
=
7
-
1
20
-
13
-
+
=
34
3+4
=
7
=
7
5
A
T
O
M
S
-
-
-
-
-
--
-
-
-
1
20
15
13
19
+
=
68
6+8
=
14
1+4
5
-
1
2
6
4
1
+
=
14
1+4
=
5
=
5
5
A
T
O
M
S
-
-
-
-
-
-
-
-
-
1
-
-
-
1
-
-
1
occurs
x
2
=
2
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
-
-
-
-
4
-
-
-
4
occurs
x
1
=
4
-
-
-
6
-
-
-
-
6
occurs
x
1
=
6
5
A
T
O
M
S
-
-
13
-
-
5
-
14
-
-
-
-
-
-
-
-
1+3
-
-
-
-
1+4
5
A
T
O
M
S
-
-
4
-
-
5
-
5
-
1
2
6
4
1
-
-
-
-
-
-
-
-
5
A
T
O
M
S
-
-
4
-
-
5
-
5

 

 

-
7
T
H
E
-
A
T
O
M
-
-
-
-
-
--
-
-
-
-
-
8
-
-
-
-
6
-
+
=
14
1+4
=
5
=
5
-
-
-
8
-
-
-
-
15
-
+
=
23
2+3
=
5
=
5
-
7
T
H
E
-
A
T
O
M
-
-
-
-
-
--
-
-
-
-
2
-
5
-
1
2
-
4
+
=
14
1+4
=
5
=
5
-
-
20
-
5
-
1
20
-
13
+
=
59
5+9
=
14
1+4
5
-
7
T
H
E
-
A
T
O
M
-
-
-
-
-
--
-
-
-
-
20
8
5
-
1
20
15
13
+
=
82
8+2
=
10
1+0
1
-
-
2
8
5
-
1
2
6
4
+
=
28
2+8
=
10
1+0
1
-
7
T
H
E
-
A
T
O
M
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
1
=
1
-
-
2
-
-
-
-
2
-
-
-
-
2
occurs
x
2
=
4
3
-
-
-
-
-
-
--
-
-
-
-
3
THREE
3
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
-
-
-
5
-
-
-
-
-
-
-
5
occurs
x
1
=
5
-
-
-
-
-
-
-
-
6
-
-
-
6
occurs
x
1
=
6
7
-
-
--
--
--
--
--
--
--
-
-
7
SEVEN
7
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
9
-
-
-
-
-
-
-
-
-
-
-
9
NINE
9
-
-
-
19
7
T
H
E
-
A
T
O
M
-
-
26
-
-
7
-
28
1+9
-
-
-
-
-
-
-
-
-
-
-
2+6
-
-
-
-
2+8
10
7
T
H
E
-
A
T
O
M
-
-
8
-
-
7
-
10
1+0
-
2
8
5
-
1
2
6
4
-
-
-
-
-
-
-
1+0
1
7
T
H
E
-
A
T
O
M
-
-
8
-
-
4
-
1

 

 

-
7
T
H
E
-
A
T
O
M
-
-
-
-
-
--
-
-
-
-
-
8
-
-
-
-
6
-
+
=
14
1+4
=
5
=
5
-
-
-
8
-
-
-
-
15
-
+
=
23
2+3
=
5
=
5
-
7
T
H
E
-
A
T
O
M
-
-
-
-
-
--
-
-
-
-
2
-
5
-
1
2
-
4
+
=
14
1+4
=
5
=
5
-
-
20
-
5
-
1
20
-
13
+
=
59
5+9
=
14
1+4
5
-
7
T
H
E
-
A
T
O
M
-
-
-
-
-
--
-
-
-
-
20
8
5
-
1
20
15
13
+
=
82
8+2
=
10
1+0
1
-
-
2
8
5
-
1
2
6
4
+
=
28
2+8
=
10
1+0
1
-
7
T
H
E
-
A
T
O
M
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
1
=
1
-
-
2
-
-
-
-
2
-
-
-
-
2
occurs
x
2
=
4
-
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
-
-
-
5
-
-
-
-
-
-
-
5
occurs
x
1
=
5
-
-
-
-
-
-
-
-
6
-
-
-
6
occurs
x
1
=
6
-
-
-
8
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
19
7
T
H
E
-
A
T
O
M
-
-
26
-
-
7
-
28
1+9
-
-
-
-
-
-
-
-
-
-
-
2+6
-
-
-
-
2+8
10
7
T
H
E
-
A
T
O
M
-
-
8
-
-
7
-
10
1+0
-
2
8
5
-
1
2
6
4
-
-
-
-
-
-
-
1+0
1
7
T
H
E
-
A
T
O
M
-
-
8
-
-
4
-
1

 

 

7
T
H
E
-
A
T
O
M
-
-
-
-
-
--
-
-
-
-
8
-
-
-
-
6
-
+
=
14
1+4
=
5
=
5
-
-
8
-
-
-
-
15
-
+
=
23
2+3
=
5
=
5
7
T
H
E
-
A
T
O
M
-
-
-
-
-
--
-
-
-
2
-
5
-
1
2
-
4
+
=
14
1+4
=
5
=
5
-
20
-
5
-
1
20
-
13
+
=
59
5+9
=
14
1+4
5
7
T
H
E
-
A
T
O
M
-
-
-
-
-
--
-
-
-
20
8
5
-
1
20
15
13
+
=
82
8+2
=
10
1+0
1
-
2
8
5
-
1
2
6
4
+
=
28
2+8
=
10
1+0
1
7
T
H
E
-
A
T
O
M
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
1
=
1
-
2
-
-
-
-
2
-
-
-
-
2
occurs
x
2
=
4
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
1
=
4
-
-
-
5
-
-
-
-
-
-
-
5
occurs
x
1
=
5
-
-
-
-
-
-
-
6
-
-
-
6
occurs
x
1
=
6
-
-
8
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
7
T
H
E
-
A
T
O
M
-
-
26
-
-
7
-
28
-
-
-
-
-
-
-
-
-
-
-
2+6
-
-
-
-
2+8
7
T
H
E
-
A
T
O
M
-
-
8
-
-
7
-
10
-
2
8
5
-
1
2
6
4
-
-
-
-
-
-
-
1+0
7
T
H
E
-
A
T
O
M
-
-
8
-
-
4
-
1

 

 

3
THE
33
15
6
6
ATOMIC
61
25
7
5
MINDS
59
23
5
1
I
9
9
9
15
Add to Reduce
162
72
27
1+5
Reduce to Deduce
1+6+2
7+2
2+7
8
Essence of Number
9
9
9

 

 

-
15
T
H
E
-
A
T
O
M
I
C
-
M
I
N
D
S
-
I
-
-
-
-
-
--
-
-
-
-
-
-
-
8
-
-
-
-
6
-
9
-
-
-
9
5
-
1
-
9
+
=
47
4+7
=
11
1+1
2
=
2
-
-
-
8
-
-
-
-
15
-
9
-
-
-
9
14
-
19
-
9
+
=
83
8+3
=
11
1+1
2
=
2
-
15
T
H
E
-
A
T
O
M
I
C
-
M
I
N
D
S
-
I
-
-
-
-
-
--
-
-
-
-
-
-
2
-
5
-
1
2
-
4
-
3
-
4
-
-
4
-
-
-
+
=
25
2+5
=
7
=
7
=
7
-
-
20
-
5
-
1
20
-
13
-
3
-
13
-
-
4
-
-
-
+
=
79
7+9
=
16
1+6
7
=
7
-
15
T
H
E
-
A
T
O
M
I
C
-
M
I
N
D
S
-
I
-
-
-
-
-
--
-
-
-
-
-
-
20
8
5
-
1
20
15
13
9
3
-
13
9
14
4
19
-
9
+
=
162
1+6+2
=
9
=
9
=
9
-
-
2
8
5
-
1
2
6
4
9
3
-
4
9
5
4
1
-
9
+
=
63
6+3
=
9
=
9
=
9
-
15
T
H
E
-
A
T
O
M
I
C
-
M
I
N
D
S
-
I
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
occurs
x
2
=
2
=
2
-
``-
2
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
-
-
-
-
-
4
-
-
-
4
-
-
4
-
-
-
-
-
4
occurs
x
3
=
12
1+2
3
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
-
6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
8
-
-
-
-
-
-
-
-
-
-
9
-
-
-
9
-
-
-
-
9
-
-
9
occurs
x
3
=
27
 2+7
9
7
15
T
H
E
-
A
T
O
M
I
C
-
M
I
N
D
S
-
I
-
-
38
-
-
15
-
63
-
36
-
1+5
-
-
-
-
-
-
-
-
9
-
-
-
9
-
-
-
-
9
-
-
3+8
-
-
1+5
-
6+3
-
3+6
7
6
T
H
E
-
A
T
O
M
I
C
-
M
I
N
D
S
-
I
-
-
11
-
-
6
-
9
-
9
-
-
2
8
5
-
1
2
6
3
9
3
-
3
9
5
4
1
-
9
-
-
1+1
-
-
-
-
-
-
-
7
6
T
H
E
-
A
T
O
M
I
C
-
M
I
N
D
S
-
I
-
-
2
-
-
6
-
9
-
9

 

 

5
T
H
E
-
A
T
O
M
I
C
-
M
I
N
D
S
-
I
-
-
-
-
-
--
-
-
-
-
-
-
8
-
-
-
-
6
-
9
-
-
-
9
5
-
1
-
9
+
=
47
4+7
=
11
1+1
2
=
2
-
-
8
-
-
-
-
15
-
9
-
-
-
9
14
-
19
-
9
+
=
83
8+3
=
11
1+1
2
=
2
15
T
H
E
-
A
T
O
M
I
C
-
M
I
N
D
S
-
I
-
-
-
-
-
--
-
-
-
-
-
2
-
5
-
1
2
-
4
-
3
-
4
-
-
4
-
-
-
+
=
25
2+5
=
7
=
7
=
7
-
20
-
5
-
1
20
-
13
-
3
-
13
-
-
4
-
-
-
+
=
79
7+9
=
16
1+6
7
=
7
15
T
H
E
-
A
T
O
M
I
C
-
M
I
N
D
S
-
I
-
-
-
-
-
--
-
-
-
-
-
20
8
5
-
1
20
15
13
9
3
-
13
9
14
4
19
-
9
+
=
162
1+6+2
=
9
=
9
=
9
-
2
8
5
-
1
2
6
4
9
3
-
4
9
5
4
1
-
9
+
=
63
6+3
=
9
=
9
=
9
15
T
H
E
-
A
T
O
M
I
C
-
M
I
N
D
S
-
I
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
occurs
x
2
=
2
=
2
``-
2
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
-
-
-
-
4
-
-
-
4
-
-
4
-
-
-
-
-
4
occurs
x
3
=
12
1+2
3
-
-
-
5
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
-
6
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
8
-
-
-
-
-
-
-
-
-
9
-
-
-
9
-
-
-
-
9
-
-
9
occurs
x
3
=
27
 2+7
9
15
T
H
E
-
A
T
O
M
I
C
-
M
I
N
D
S
-
I
-
-
38
-
-
15
-
63
-
36
1+5
-
-
-
-
-
-
-
-
9
-
-
-
9
-
-
-
-
9
-
-
3+8
-
-
1+5
-
6+3
-
3+6
6
T
H
E
-
A
T
O
M
I
C
-
M
I
N
D
S
-
I
-
-
11
-
-
6
-
9
-
9
-
2
8
5
-
1
2
6
3
9
3
-
3
9
5
4
1
-
9
-
-
1+1
-
-
-
-
-
-
-
6
T
H
E
-
A
T
O
M
I
C
-
M
I
N
D
S
-
I
-
-
2
-
-
6
-
9
-
9

 

 

5
T
H
E
A
T
O
M
I
C
M
I
N
D
S
I
-
-
-
-
-
--
-
-
-
-
-
-
8
-
-
-
6
-
9
-
-
9
5
-
1
9
+
=
47
4+7
=
11
1+1
2
=
2
-
-
8
-
-
-
15
-
9
-
-
9
14
-
19
9
+
=
83
8+3
=
11
1+1
2
=
2
15
T
H
E
A
T
O
M
I
C
M
I
N
D
S
I
-
-
-
-
-
--
-
-
-
-
-
2
-
5
1
2
-
4
-
3
4
-
-
4
-
-
+
=
25
2+5
=
7
=
7
=
7
-
20
-
5
1
20
-
13
-
3
13
-
-
4
-
-
+
=
79
7+9
=
16
1+6
7
=
7
15
T
H
E
A
T
O
M
I
C
M
I
N
D
S
I
-
-
-
-
-
--
-
-
-
-
-
20
8
5
1
20
15
13
9
3
13
9
14
4
19
9
+
=
162
1+6+2
=
9
=
9
=
9
-
2
8
5
1
2
6
4
9
3
4
9
5
4
1
9
+
=
63
6+3
=
9
=
9
=
9
15
T
H
E
A
T
O
M
I
C
M
I
N
D
S
I
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
1
-
-
-
1
occurs
x
2
=
2
=
2
``-
2
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
-
-
-
4
-
-
4
-
-
4
-
-
-
-
4
occurs
x
3
=
12
1+2
3
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
-
6
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
8
-
-
-
-
-
-
-
-
9
-
-
9
-
-
-
9
-
-
9
occurs
x
3
=
27
 2+7
9
15
T
H
E
A
T
O
M
I
C
M
I
N
D
S
I
-
-
38
-
-
15
-
63
-
36
1+5
-
-
-
-
-
-
-
9
-
-
9
-
-
-
9
-
-
3+8
-
-
1+5
-
6+3
-
3+6
6
T
H
E
A
T
O
M
I
C
M
I
N
D
S
I
-
-
11
-
-
6
-
9
-
9
-
2
8
5
1
2
6
3
9
3
3
9
5
4
1
9
-
-
1+1
-
-
-
-
-
-
-
6
T
H
E
A
T
O
M
I
C
M
I
N
D
S
I
-
-
2
-
-
6
-
9
-
9

 

 

D
=
4
4
DARK
34
16
7
E
=
5
6
ENERGY
74
38
2
-
-
9
10
Add to Reduce
108
54
9
-
=
-
1+0
Reduce to Deduce
1+0+8
5+4
-
-
-
9
1
Essence of Number
9
9
9

 

 

-
10
D
A
R
K
-
E
N
E
R
G
Y
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
+
=
5
-
=
5
=
5
=
5
-
-
-
-
-
-
-
-
14
-
-
-
-
+
=
14
1+4
=
5
=
5
=
5
-
10
D
A
R
K
-
E
N
E
R
G
Y
-
-
-
-
-
-
-
-
-
-
-
-
4
1
9
2
-
5
-
5
9
7
7
+
=
49
4+9
=
13
1+3
4
=
4
-
-
4
1
18
11
-
5
-
5
18
7
25
+
=
94
9+4
=
13
1+3
4
=
4
-
10
D
A
R
K
-
E
N
E
R
G
Y
-
-
-
-
-
-
-
-
-
-
-
-
4
1
18
11
-
5
14
5
18
7
25
+
=
108
1+0+8
=
9
=
9
=
9
-
-
4
1
9
2
-
5
5
5
9
7
7
+
=
54
5+4
=
9
=
9
=
9
-
10
D
A
R
K
-
E
N
E
R
G
Y
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
THREE
3
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
5
5
5
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
-
--
-
--
-
-
--
-
-
-
-
7
7
-
-
7
occurs
x
2
=
14
1+4
5
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
9
-
-
-
-
9
occurs
x
2
=
18
1+8
9
17
10
D
A
R
K
-
E
N
E
R
G
Y
-
-
28
-
-
10
-
54
-
27
1+7
1+0
-
-
-
-
-
5
5
5
-
-
-
-
-
2+8
-
-
1+0
-
5+4
-
2+7
8
1
D
A
R
K
-
E
N
E
R
G
Y
-
-
10
-
-
1
-
9
-
9
-
--
4
1
9
2
-
5
5
5
9
7
7
-
-
1+0
-
-
--
-
-
-
-
8
1
D
A
R
K
-
E
N
E
R
G
Y
-
-
1
-
-
1
-
9
-
9

 

 

10
D
A
R
K
-
E
N
E
R
G
Y
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
+
=
5
-
=
5
=
5
=
5
-
-
-
-
-
-
-
14
-
-
-
-
+
=
14
1+4
=
5
=
5
=
5
10
D
A
R
K
-
E
N
E
R
G
Y
-
-
-
-
-
-
-
-
-
-
-
4
1
9
2
-
5
-
5
9
7
7
+
=
49
4+9
=
13
1+3
4
=
4
-
4
1
18
11
-
5
-
5
18
7
25
+
=
94
9+4
=
13
1+3
4
=
4
10
D
A
R
K
-
E
N
E
R
G
Y
-
-
-
-
-
-
-
-
-
-
-
4
1
18
11
-
5
14
5
18
7
25
+
=
108
1+0+8
=
9
=
9
=
9
-
4
1
9
2
-
5
5
5
9
7
7
+
=
54
5+4
=
9
=
9
=
9
10
D
A
R
K
-
E
N
E
R
G
Y
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
-
2
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
4
-
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
5
5
5
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
--
-
--
-
-
--
-
-
-
-
7
7
-
-
7
occurs
x
2
=
14
1+4
5
-
-
-
9
-
-
-
-
-
9
-
-
-
-
9
occurs
x
2
=
18
1+8
9
10
D
A
R
K
-
E
N
E
R
G
Y
-
-
28
-
-
10
-
54
-
27
1+0
-
-
-
-
-
5
5
5
-
-
-
-
-
2+8
-
-
1+0
-
5+4
-
2+7
1
D
A
R
K
-
E
N
E
R
G
Y
-
-
10
-
-
1
-
9
-
9
--
4
1
9
2
-
5
5
5
9
7
7
-
-
1+0
-
-
--
-
-
-
-
1
D
A
R
K
-
E
N
E
R
G
Y
-
-
1
-
-
1
-
9
-
9

 

 

10
D
A
R
K
E
N
E
R
G
Y
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
+
=
5
-
=
5
=
5
=
5
-
-
-
-
-
-
14
-
-
-
-
+
=
14
1+4
=
5
=
5
=
5
10
D
A
R
K
E
N
E
R
G
Y
-
-
-
-
-
-
-
-
-
-
-
4
1
9
2
5
-
5
9
7
7
+
=
49
4+9
=
13
1+3
4
=
4
-
4
1
18
11
5
-
5
18
7
25
+
=
94
9+4
=
13
1+3
4
=
4
10
D
A
R
K
E
N
E
R
G
Y
-
-
-
-
-
-
-
-
-
-
-
4
1
18
11
5
14
5
18
7
25
+
=
108
1+0+8
=
9
=
9
=
9
-
4
1
9
2
5
5
5
9
7
7
+
=
54
5+4
=
9
=
9
=
9
10
D
A
R
K
E
N
E
R
G
Y
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
-
2
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
4
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
5
5
5
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
--
-
--
-
-
-
-
-
-
7
7
-
-
7
occurs
x
2
=
14
1+4
5
-
-
-
9
-
-
-
-
9
-
-
-
-
9
occurs
x
2
=
18
1+8
9
10
D
A
R
K
E
N
E
R
G
Y
-
-
28
-
-
10
-
54
-
27
1+0
-
-
-
-
5
5
5
-
-
-
-
-
2+8
-
-
1+0
-
5+4
-
2+7
1
D
A
R
K
E
N
E
R
G
Y
-
-
10
-
-
1
-
9
-
9
--
4
1
9
2
5
5
5
9
7
7
-
-
1+0
-
-
--
-
-
-
-
1
D
A
R
K
E
N
E
R
G
Y
-
-
1
-
-
1
-
9
-
9

 

 

-
4
C
E
L
L
-
-
-
-
-
--
-
-
-
`-
3
5
12
12
+
=
32
3+2
=
5
=
5
-
-
3
5
3
3
+
=
14
1+4
=
5
=
5
-
4
C
E
L
L
-
-
-
-
-
--
-
-
-
`-
3
5
12
12
+
=
32
3+2
=
5
=
5
-
-
3
5
3
3
+
=
14
1+4
=
5
=
5
-
4
C
E
L
L
T
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
ONE
1
-
-
-
-2-
-
-
-
-
-
-
-
2
TWO
2
-
-
-
-
-
3
-
3
3
-
-
3
occurs
x
3
=
9
4
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
--
-
5
-
-
-
-
5
occurs
x
1
=
5
6
-
-
-
-
-
-
-
6
SIX
6
-
-
-
7
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
8
--
-
-
-
-
-
-
8
EIGHT
8
-
-
-
9
-
-
-
-
-
-
-
9
NINE
9
-
-
-
37
4
C
E
L
L
-
-
8
-
-
4
-
14
3+7
-
3
-
3
3
-
-
-
-
-
-
-
1+4
10
4
C
E
L
L
-
-
8
-
-
4
-
5
1+0
-
3
5
3
3
-
-
-
-
-
-
-
-
1
4
C
E
L
L
-
-
8
-
-
4
-
5

 

 

4
C
E
L
L
-
-
-
-
-
--
-
-
`-
3
5
12
12
+
=
32
3+2
=
5
=
5
-
3
5
3
3
+
=
14
1+4
=
5
=
5
4
C
E
L
L
-
-
-
-
-
--
-
-
`-
3
5
12
12
+
=
32
3+2
=
5
=
5
-
3
5
3
3
+
=
14
1+4
=
5
=
5
4
C
E
L
L
T
-
-
-
-
-
-
-
-
3
-
3
3
-
-
3
occurs
x
3
=
9
--
-
5
-
-
-
-
5
occurs
x
1
=
5
4
C
E
L
L
-
-
8
-
-
4
-
14
-
3
-
3
3
-
-
-
-
-
-
-
1+4
4
C
E
L
L
-
-
8
-
-
4
-
5
-
3
5
3
3
-
-
-
-
-
-
-
-
4
C
E
L
L
-
-
8
-
-
4
-
5

 

 

-
5
C
E
L
L
S
-
-
-
-
-
--
-
-
-
-
-
-
-
-
1
+
=
1
-
=
1
=
1
-
`-
-
-
-
-
19
+
=
19
1+9
=
10
1+0
1
-
5
C
E
L
L
S
-
-
-
-
-
--
-
-
-
-
3
5
3
3
-
+
=
14
1+4
=
5
=
5
-
`-
3
5
12
12
-
+
=
32
3+2
=
5
=
5
-
5
C
E
L
L
S
-
-
-
-
-
--
-
-
-
`-
3
5
12
12
19
+
=
51
5+1
=
6
=
6
-
-
3
5
3
3
1
+
=
15
1+5
=
6
=
6
-
5
C
E
L
L
S
T
-
-
-
-
-
-
-
--
-
-
-
-
-
1
-
-
1
occurs
x
1
=
1
-2-
-
-
-
-
-
-
-
-
2
TWO
2
-
-
-
-
-
3
-
3
3
-
-
-
3
occurs
x
3
=
9
4
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
--
-
5
-
-
-
-
-
5
occurs
x
1
=
5
6
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
7
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
8
--
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
9
-
-
-
-
-
-
-
-
9
NINE
9
-
-
-
36
5
C
E
L
L
S
-
-
9
-
-
5
-
15
3+6
-
3
-
3
3
-
-
-
-
-
-
-
-
1+5
9
5
C
E
L
L
S
-
-
9
-
-
5
-
6
-
-
3
5
3
3
1
-
-
-
-
-
-
-
-
9
5
C
E
L
L
S
-
-
9
-
-
5
-
6

 

 

5
C
E
L
L
S
-
-
-
-
-
--
-
-
-
-
-
-
-
1
+
=
1
-
=
1
=
1
`-
-
-
-
-
19
+
=
19
1+9
=
10
1+0
1
5
C
E
L
L
S
-
-
-
-
-
--
-
-
-
3
5
3
3
-
+
=
14
1+4
=
5
=
5
`-
3
5
12
12
-
+
=
32
3+2
=
5
=
5
5
C
E
L
L
S
-
-
-
-
-
--
-
-
`-
3
5
12
12
19
+
=
51
5+1
=
6
=
6
-
3
5
3
3
1
+
=
15
1+5
=
6
=
6
5
C
E
L
L
S
T
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
1
=
1
-
3
-
3
3
-
-
-
3
occurs
x
3
=
9
--
-
5
-
-
-
-
-
5
occurs
x
1
=
5
5
C
E
L
L
S
-
-
9
-
-
5
-
15
-
3
-
3
3
-
-
-
-
-
-
-
-
1+5
5
C
E
L
L
S
-
-
9
-
-
5
-
6
-
3
5
3
3
1
-
-
-
-
-
-
-
-
5
C
E
L
L
S
-
-
9
-
-
5
-
6

 

 

6
ATOMIC
61
25
7
6
THEORY
76
31
4
12
First Total
137
56
11
1+2
Add to Reduce
1+3+7
5+6
1+1
3
Second Total
11
11
2
-
Reduce to Deduce
1+1
1+1
-
3
Essence of Number
2
2
2

 

 

3
THE
33
15
6
8
STANDARD
81
27
9
5
MODEL
49
22
4
16
First Total
163
64
19
1+6
Add to Reduce
1+6+3
6+4
1+9
7
Second Total
10
10
10
-
Reduce to Deduce
1+0
1+0
1+0
7
Essence of Number
1
1
1

 

 

-
-
-
-T
-
LIGHT
-
-
-
-
-
-
-
4
LIGH
36
27
9
-
-
-
-
1
T
20
2
2
L
=
3
-
5
LIGHT
56
29
11
-
-
-
-
-
-
5+6
2+9
1+1
-
-
1
-
9
-
11
11
2
-
-
-
-
-
-
1+1
1+1
-
-
-
1
-
9
-
2
2
2

 

 

-T
5
L
I
G
H
T
-
-
-
-
-
--
-
-
-
-
-
`-
-
9
-
8
-
+
=
17
1+7
=
8
=
8
-
8
-
5
L
I
G
H
T
-
-
-
-
-
--
-
-
-
-
-
-
3
-
7
-
2
+
=
12
1+2
=
3
=
3
-
3
-
`-
12
-
7
-
20
+
=
39
3+9
=
12
1+2
3
-
3
-
5
L
I
G
H
T
-
-
-
-
-
--
-
-
-
-
-
`-
12
9
7
8
20
+
=
56
5+6
=
11
1+1
2
-
2
-
-
3
9
7
8
2
+
=
29
2+9
=
11
1+1
2
-
2
-
5
L
I
G
H
T
-T
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
2
occurs
x
1
=
2
=
2
-
-
3
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
4
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
--
--
-
-
7
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
-
-
-
8
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
9
-
-
-
-
-
9
occurs
x
1
=
9
=
9
16
5
L
I
G
H
T
-
-
29
-
1
5
-
29
-
29
1+6
-
-
9
-
-
-
-
-
2+9
-
-
-
-
2+9
-
2+9
7
5
L
I
G
H
T
-
-
11
-
1
5
-
11
-
11
-
-
3
9
7
8
2
-
-
1+1
-
-
-
-
1+1
-
1+1
7
5
L
I
G
H
T
-
-
2
-
-
5
-
2
-
2

 

 

5
L
I
G
H
T
-
-
-
-
-
--
-
-
-
-
`-
-
9
-
8
-
+
=
17
1+7
=
8
=
8
-
8
5
L
I
G
H
T
-
-
-
-
-
--
-
-
-
-
-
3
-
7
-
2
+
=
12
1+2
=
3
=
3
-
3
`-
12
-
7
-
20
+
=
39
3+9
=
12
1+2
3
-
3
5
L
I
G
H
T
-
-
-
-
-
--
-
-
-
-
`-
12
9
7
8
20
+
=
56
5+6
=
11
1+1
2
-
2
-
3
9
7
8
2
+
=
29
2+9
=
11
1+1
2
-
2
5
L
I
G
H
T
-T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
2
occurs
x
1
=
2
=
2
-
3
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
7
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
-
-
8
-
-
-
8
occurs
x
1
=
8
=
8
-
-
9
-
-
-
-
-
9
occurs
x
1
=
9
=
9
5
L
I
G
H
T
-
-
29
-
1
5
-
29
-
29
-
-
9
-
-
-
-
-
2+9
-
-
-
-
2+9
-
2+9
5
L
I
G
H
T
-
-
11
-
1
5
-
11
-
11
-
3
9
7
8
2
-
-
1+1
-
-
-
-
1+1
-
1+1
5
L
I
G
H
T
-
-
2
-
-
5
-
2
-
2

 

 

-
10
L
I
G
H
T
-
W
A
V
E
S
-
-
-
-
-
--
-
-
-
-
-
`-
-
9
-
8
-
-
-
-
-
-
1
+
=
18
1+8
=
9
-
9
-
9
-
`-
-
9
-
8
-
-
-
-
-
-
19
+
=
36
3+6
=
9
-
9
-
9
-
10
L
I
G
H
T
-
W
A
V
E
S
-
-
-
-
-
--
-
-
-
-
-
-
3
-
7
-
2
-
5
1
4
5
-
+
=
27
2+7
=
9
-
9
-
9
-
`-
12
-
7
-
20
-
23
1
22
5
-
+
=
90
7+5
=
9
-
9
-
9
-
10
L
I
G
H
T
-
W
A
V
E
S
-T
-
-
-
-
-
-
-
-
-
-
`-
12
9
7
8
20
-
23
1
22
5
19
+
=
126
1+2+6
=
9
-
9
-
9
-T
-
3
9
7
8
2
-T
5
1
4
5
1
+
=
45
4+5
=
9
-
9
-
9
-
10
L
I
G
H
T
-
W
A
V
E
S
-T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
-
5
-
-
5
-
-
-
5
occurs
x
2
=
10
1+0
1
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
-
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
6
10
L
I
G
H
T
-
W
A
V
E
S
-
-
39
-
1
12
-
45
-
36
-
1+0
-
9
-
-
-
-
-
-
-
-
-
-
-
3+9
-
-
1+2
-
4+5
-
3+6
6
1
L
I
G
H
T
-
W
A
V
E
S
-
-
12
-
1
3
-
9
-
9
-
-
3
9
7
8
2
-
5
1
4
5
1
-
-
1+2
-
-
-
-
-
-
-
6
1
L
I
G
H
T
-
W
A
V
E
S
-
-
3
-
-
3
-
9
-
9

 

 

5
WAVES
70
16
7
9
PARTICLES
103
40
4

 

 

4
WAVE
51
15
6
9
PARTICLES
103
40
4
13
First Total
154
55
10
1+3
Add to Reduce
1+5+4
5+5
1+0
4
Second Total
10
10
1
-
Reduce to Deduce
1+0
1+0
-
4
Essence of Number
1
1
1

 

 

W
=
5
4
WAVE
51
15
6
P
=
7
8
PARTICLE
84
39
3
-
-
12
12
Add to Reduce
135
54
9
-
-
1+2
1+2
Reduce to Deduce
1+3+5
5+4
-
-
-
3
3
Essence of Number
9
9
9

 

 

-
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
+
=
9
-
=
9
-
9
-
9
-
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
--
-
-
-
-
-
-
5
1
4
5
-
7
1
9
2
-
3
3
5
+
=
45
4+5
=
9
-
9
-
9
-
-
23
1
22
5
-
16
1
18
20
-
3
12
5
+
=
126
1+2+6
=
9
-
9
-
9
-
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
--
-
-
-
-
-
-
23
1
22
5
-
16
1
18
20
9
3
12
5
+
=
135
1+3+5
=
9
-
9
=
9
-
-
5
1
4
5
-
7
1
9
2
9
3
3
5
+
=
54
5+4
=
9
-
9
=
9
-
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
-
2
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
-
-
-
-
-
-
-
-
3
3
-
-
-
3
occurs
x
2
=
6
-
6
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
-
4
-
-
5
-
-
5
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
3
=
15
1+5
6
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
-
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
9
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
14
12
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
31
-
-
12
-
54
-
36
1+4
1+2
-
-
-
-
-
-
-
9
-
9
-
-
-
-
-
3+1
-
-
1+2
-
5+4
-
3+6
5
3
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
4
-
-
3
-
9
-
9
-
-
5
1
4
5
-
7
1
9
2
9
3
3
5
-
-
-
-
-
-
-
-
-
-
5
3
W
A
V
E
-
P
A
R
T
I
C
L
E
-
-
4
-
-
3
-
9
-
9

 

 

P
=
7
-
4
PARTICLES
103
40
4
O
=
6
-
1
OF
21
12
3
L
=
3
-
5
LIGHT
56
29
2
-
-
16
-
10
-
180
81
9
-
-
1+6
-
1+0
-
1+8+0
8+1
-
-
-
7
-
1
-
9
9
9

 

 

-T
16
P
A
R
T
I
C
L
E
S
-
O
F
-
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-T
`-
-
-
-
-
9
-
-
-
1
-
6
-
-
-
9
-
8
-
+
=
33
3+3
=
6
=
6
=
6
-
`-
-
-
-
-
9
-
-
-
19
-
15
-
-
-
9
-
8
-
+
=
60
6+0
=
6
=
6
=
6
-
16
P
A
R
T
I
C
L
E
S
-
O
F
-
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
-
7
1
9
2
-
3
3
5
-
-
-
6
-
3
-
7
-
2
+
=
48
4+8
=
12
1+2
3
=
3
-
`-
16
1
18
20
-
3
12
5
-
-
-
6
-
12
-
7
-
20
+
=
120
1+2+0
=
3
=
3
=
3
-
16
P
A
R
T
I
C
L
E
S
-
O
F
-
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
`-
16
1
18
20
9
3
12
5
19
-
15
6
-
12
9
7
8
20
+
=
180
1+8+0
=
9
=
9
=
9
-
-
7
1
9
2
9
3
3
5
1
-T
6
6
-T
3
9
7
8
2
+
=
81
8+1
=
9
=
9
=
9
-
16
P
A
R
T
I
C
L
E
S
-
O
F
-
L
I
G
H
T
-T
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
-
3
3
-
-
-
-
-
-
3
-
-
-
-
-
-
3
occurs
x
3
=
9
=
9
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
-
5
occurs
x
1
=
5
=
5
-
-
-
-
-
-
-
-
-
-
-
-
6
6
-
-
-
-
-
-
-
-
6
occurs
x
2
=
12
1+2
3
-
--
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
7
occurs
x
2
=
14
1+4
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
9
-
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
9
occurs
x
3
=
27
=
9
4
16
P
A
R
T
I
C
L
E
S
-
O
F
-
L
I
G
H
T
-
-
41
-
1
16
-
81
-
45
-
-
-
-
9
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
4+1
-
-
1+6
-
8+1
-
4+5
4
16
P
A
R
T
I
C
L
E
S
-
O
F
-
L
I
G
H
T
-
-
5
-
1
7
-
9
-
9
-
-
7
1
9
2
9
3
-
5
1
-
6
6
-
3
9
7
8
2
-
-
-
-
-
-
-
-
-
-
4
16
P
A
R
T
I
C
L
E
S
-
O
F
-
L
I
G
H
T
-
-
5
-
-
7
-
9
-
9

 

 

T
=
2
-
3
THE
33
15
6
L
=
3
-
5
LARGE
43
25
7
H
=
8
-
6
HADRON
60
33
6
C
=
3
-
8
COLLIDER
78
42
6
-
-
16
-
22
Add to Reduce
214
115
25
-
-
1+6
-
2+2
Reduce to Deduce
2+1+4
1+1+5
2+5
-
-
7
-
4
Essence of Number
7
7
7

 

 

21-
22
T
H
E
-
L
A
R
G
E
-
H
A
D
R
O
N
-
C
O
L
L
I
D
E
R
-
-
-
-
-
--
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
8
-
-
-
6
5
-
-
6
-
-
9
-
-
-
+
=
42
4+2
=
6
=
6
=
6
-
-
-
8
-
-
-
-
-
-
-
-
8
-
-
-
15
14
-
-
15
-
-
9
-
-
-
+
=
69
6+9
=
15
1+5
6
=
6
-
22
T
H
E
-
L
A
R
G
E
-
H
A
D
R
O
N
-
C
O
L
L
I
D
E
R
-
-
-
-
-
-
-
-
-
-
-
-
2
-
5
-
3
1
9
7
5
-
-
1
4
9
-
-
-
3
-
3
3
-
4
5
9
+
=
73
7+3
=
10
1+0
1
=
1
-
-
20
-
5
-
12
1
18
7
5
-
-
1
4
18
-
-
-
3
-
12
12
-
4
5
18
+
=
145
1+4+5
=
10
1+0
1
=
1
-
22
T
H
E
-
L
A
R
G
E
-
H
A
D
R
O
N
-
C
O
L
L
I
D
E
R
-
-
-
-
-
--
-
-
-
-
-
-
20
8
5
-
12
1
18
7
5
-
8
1
4
18
15
14
-
3
15
12
12
9
4
5
18
+
=
214
2+1+4
=
7
=
7
=
7
-
-
2
8
5
-
3
1
9
7
5
-
8
1
4
9
6
5
-
3
6
3
3
9
4
5
9
+
=
115
1+1+5
=
7
=
7
=
7
-
22
T
H
E
-
L
A
R
G
E
-
H
A
D
R
O
N
-
C
O
L
L
I
D
E
R
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
2
``-
2
-
-
-`
-
-
-
-
-
-`
-
-
-
-
-
-
-`
-
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
3
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
3
-
3
3
-
-
-
-
-
-
3
occurs
x
4
=
12
1+2
3
4
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
4
-
-
-
-
4
occurs
x
2
=
8
=
8
5
-
-
-
5
-
-
-
-
-
5
-
-
-
-
-
-
5
-
-
-
-
-
-
-
5
-
-
-
5
occurs
x
4
=
20
2+0
2
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
6
-
-
-
-
-
-
-
-
6
occurs
x
2
=
12
1+2
3
7
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
8
-
-
8
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
2
=
16
=
7
9
-
-
-
-
-
-
-
9
-
-
-
-
-
-
9
-
-
-
-
-
-
-
9
-
-
9
-
-
9
occurs
x
4
=
36
3+6
9
45
22
T
H
E
-
L
A
R
G
E
-
H
A
D
R
O
N
-
C
O
L
L
I
D
E
R
-
-
45
-
-
22
-
86
-
43
4+5
2+2
-
-
-
-
-
-
9
-
-
-
-
-
-
9
-
-
-
-
-
-
-
9
-
-
9
-
-
4+5
-
-
2+2
-
8+6
-
4+3
9
4
T
H
E
-
L
A
R
G
E
-
H
A
D
R
O
N
-
C
O
L
L
I
D
E
R
-
-
9
-
-
4
-
14
-
7
-
-
2
8
5
-
3
1
9
7
5
-
8
1
4
9
6
5
-
3
6
3
3
9
4
5
9
-
-
-
-
-
-
-
1+4
-
-
9
4
T
H
E
-
L
A
R
G
E
-
H
A
D
R
O
N
-
C
O
L
L
I
D
E
R
-
-
9
-
-
4
-
5
-
7

 

 

L
=
3
-
5
LARGE
43
25
7
H
=
8
-
6
HADRON
60
33
6
C
=
3
-
8
COLLIDER
78
42
6
-
-
14
-
19
First Total
181
100
19
-
-
1+4
-
1+9
Add to Reduce
1+8+1
1+0+0
1+9
-
-
5
-
10
Second Total
10
1
10
-
-
-
-
1+0
Reduce to Deduce
1+0
-
1+0
-
-
5
-
1
Essence of Number
1
1
1

 

 

21-
19
L
A
R
G
E
-
H
A
D
R
O
N
-
C
O
L
L
I
D
E
R
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
6
5
-
-
6
-
-
9
-
-
-
+
=
34
3+4
=
7
=
7
=
7
-
-
-
-
-
-
-
-
8
-
-
-
15
14
-
-
15
-
-
9
-
-
-
+
=
61
6+1
=
7
=
7
=
7
-
19
L
A
R
G
E
-
H
A
D
R
O
N
-
C
O
L
L
I
D
E
R
-
-
-
-
-
-
-
-
-
-
-
-
3
1
9
7
5
-
-
1
4
9
-
-
-
3
-
3
3
-
4
5
9
+
=
66
6+6
=
12
1+2
3
=
3
-
-
12
1
18
7
5
-
-
1
4
18
-
-
-
3
-
12
12
-
4
5
18
+
=
120
1+2+0
=
3
=
3
=
3
-
19
L
A
R
G
E
-
H
A
D
R
O
N
-
C
O
L
L
I
D
E
R
-
-
-
-
-
--
-
-
-
-
-
-
12
1
18
7
5
-
8
1
4
18
15
14
-
3
15
12
12
9
4
5
18
+
=
181
1+8+1
=
10
1+0
1
=
1
-
-
3
1
9
7
5
-
8
1
4
9
6
5
-
3
6
3
3
9
4
5
9
+
=
100
1+0+0
=
1
=
1
=
1
-
19
L
A
R
G
E
-
H
A
D
R
O
N
-
C
O
L
L
I
D
E
R
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
2
``-
-
-
-
-
-
-`
-
-
-
-
-
-
-`
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
3
-
3
3
-
-
-
-
-
-
3
occurs
x
4
=
12
1+2
3
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
4
-
-
-
-
4
occurs
x
2
=
8
=
8
-
-
-
-
-
-
5
-
-
-
-
-
-
5
-
-
-
-
-
-
-
5
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
6
-
-
-
-
-
-
-
-
6
occurs
x
2
=
12
1+2
3
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
9
-
-
-
-
-
-
9
-
-
-
-
-
-
-
9
-
-
9
-
-
9
occurs
x
4
=
36
3+6
9
2
19
L
A
R
G
E
-
H
A
D
R
O
N
-
C
O
L
L
I
D
E
R
-
-
43
-
-
19
-
100
-
46
-
1+9
-
-
9
-
-
-
-
-
-
9
-
-
-
-
-
-
-
9
-
-
9
-
-
4+3
-
-
1+9
-
1+0+0
-
4+6
2
10
L
A
R
G
E
-
H
A
D
R
O
N
-
C
O
L
L
I
D
E
R
-
-
7
-
-
10
-
1
-
10
-
1+0
3
1
9
7
5
-
8
1
4
9
6
5
-
3
6
3
3
9
4
5
9
-
-
-
-
-
1+0
-
-
-
1+0
2
1
L
A
R
G
E
-
H
A
D
R
O
N
-
C
O
L
L
I
D
E
R
-
-
7
-
-
1
-
1
-
1

 

 

T
=
2
-
3
THE
33
15
6
H
=
8
-
6
HADRON
60
33
6
C
=
3
-
8
COLLIDER
78
42
6
-
-
13
-
17
Add to Reduce
171
90
18
-
-
1+3
-
1+7
Reduce to Deduce
1+7+1
1+1+5
2+5
-
-
4
-
8
Essence of Number
9
9
9

 

 

-
17
T
H
E
-
H
A
D
R
O
N
-
C
O
L
L
I
D
E
R
-
-
-
-
-
--
-
-
-
-
-
-
-
8
-
-
8
-
-
-
6
5
-
-
6
-
-
9
-
-
-
+
=
42
4+2
=
6
=
6
=
6
-
-
-
8
-
-
8
-
-
-
15
14
-
-
15
-
-
9
-
-
-
+
=
69
6+9
=
15
1+5
6
=
6
-
17
T
H
E
-
H
A
D
R
O
N
-
C
O
L
L
I
D
E
R
-
-
-
-
-
-
-
-
-
-
-
-
2
-
5
-
-
1
4
9
-
-
-
3
-
3
3
-
4
5
9
+
=
48
7+3
=
10
=
3
=
3
-
-
20
-
5
-
-
1
4
18
-
-
-
3
-
12
12
-
4
5
18
+
=
102
1+0+2
=
3
=
3
=
3
-
17
T
H
E
-
H
A
D
R
O
N
-
C
O
L
L
I
D
E
R
-
-
-
-
-
--
-
-
-
-
-
-
20
8
5
-
8
1
4
18
15
14
-
3
15
12
12
9
4
5
18
+
=
171
1+7+1
=
9
=
9
=
9
-
-
2
8
5
-
8
1
4
9
6
5
-
3
6
3
3
9
4
5
9
+
=
90
9+0
=
9
=
9
=
9
-
17
T
H
E
-
H
A
D
R
O
N
-
C
O
L
L
I
D
E
R
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
``-
2
-
-
-`
-
-
-
-
-
-
-`
-
-
-
-
-
-
-
-
-
-
2
occurs
x
1
=
2
=
2
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
3
3
-
-
-
-
-
-
3
occurs
x
3
=
9
=
9
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
4
-
-
-
-
4
occurs
x
2
=
8
=
8
-
-
-
-
5
-
-
-
-
-
-
5
-
-
-
-
-
-
-
5
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
-
-
-
-
6
-
-
-
6
-
-
-
-
-
-
-
-
6
occurs
x
2
=
12
1+2
3
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
8
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
2
=
16
=
7
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
9
-
-
9
-
-
9
occurs
x
3
=
27
2+7
9
7
17
T
H
E
-
H
A
D
R
O
N
-
C
O
L
L
I
D
E
R
-
-
45
-
-
17
-
90
-
45
-
1+7
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
9
-
-
9
-
-
4+5
-
-
1+7
-
9+0
-
4+5
7
8
T
H
E
-
H
A
D
R
O
N
-
C
O
L
L
I
D
E
R
-
-
9
-
-
8
-
9
-
9
-
-
2
8
5
-
8
1
4
9
6
5
-
3
6
3
3
9
4
5
9
-
-
-
-
-
-
-
-
-
-
7
8
T
H
E
-
H
A
D
R
O
N
-
C
O
L
L
I
D
E
R
-
-
9
-
-
8
-
9
-
9

 

 

P
=
7
-
8
PARTICLE
84
39
3
A
=
1
-
11
ACCELERATOR
101
47
2
-
-
8
-
19
First Total
185
86
5
-
-
-
-
1+9
Add to Reduce
1+8+5
8+6
-
-
-
8
-
10
Second Total
14
14
5
-
-
-
-
1+0
Reduce to Deduce
1+4
1+4
-
-
-
8
-
1
Essence of Number
5
5
5

 

 

-T
19
P
A
R
T
I
C
L
E
-
A
C
C
E
L
E
R
A
T
O
R
-
-
-
-
-
--
-
-
-
-
-T
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
+
=
15
1+5
=
6
=
6
=
6
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
15
-
+
=
24
2+4
=
6
=
6
=
6
-
19
P
A
R
T
I
C
L
E
-
A
C
C
E
L
E
R
A
T
O
R
-
-
-
-
-
--
-
-
-
-
-
-
7
1
9
2
-
3
3
5
-
1
3
3
5
3
5
9
1
2
-
9
+
=
71
7+1
=
8
=
8
=
8
-
-
16
1
18
20
-
3
12
5
-
1
3
3
5
12
5
18
1
20
-
18
+
=
161
1+6+1
=
8
=
8
=
8
-
19
P
A
R
T
I
C
L
E
-
A
C
C
E
L
E
R
A
T
O
R
-
-
-
-
-
--
-
-
-
-
-
-
16
1
18
20
9
3
12
5
-
1
3
3
5
12
5
18
1
20
15
18
+
=
185
1+8+5
=
14
1+4
5
=
5
-
-
7
1
9
2
9
3
3
5
-T
1
3
3
5
3
5
9
1
2
6
9
+
=
86
8+6
=
14
1+4
5
=
5
-
19
P
A
R
T
I
C
L
E
-
A
C
C
E
L
E
R
A
T
O
R
-T
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
-
-
-
-
-
-
1
-
-
-
-
-
1
occurs
x
3
=
3
=
3
-
``-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
-
3
3
-
-
-
3
3
-
3
-
-
-
-
-
-
-
-
3
occurs
x
5
=
15
1+5
6
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
5
-
5
-
-
-
-
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
6
occurs
x
1
=
6
=
6
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
=
7
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
9
-
9
-
-
-
-
-
-
-
-
-
-
9
-
-
-
9
-
-
9
occurs
x
3
=
27
=
9
12
19
P
A
R
T
I
C
L
E
-
A
C
C
E
L
E
R
A
T
O
R
-
-
33
-
1
19
-
86
-
41
1+2
1+9
-
-
9
-
9
-
-
-
-
-
-
-
-
-
-
9
-
-
-
9
-
-
3+3
-
-
1+9
-
8+6
-
4+1
3
10
P
A
R
T
I
C
L
E
-
A
C
C
E
L
E
R
A
T
O
R
-
-
6
-
1
10
-
14
-
5
-
1+0
7
1
9
2
9
3
-
5
-
1
3
3
5
-
5
9
1
2
6
9
-
-
-
-
-
1+0
-
1+4
-
-
3
1
P
A
R
T
I
C
L
E
-
A
C
C
E
L
E
R
A
T
O
R
-
-
6
-
-
1
-
5
-
5

 

 

P
=
7
-
8
PARTICLE
84
39
3
C
=
3
-
8
COLLIDER
78
42
6
-
-
10
-
16
Add to Reduce
162
81
9
-
-
1+0
-
1+6
Reduce to Deduce
1+6+2
8+6
-
-
-
1
-
7
Essence of Number
9
9
9

 

 

3
THE
33
15
6
4
BEAM
21
12
3
7
Add to Reduce
54
27
9
-
Reduce to Deduce
5+4
2+7
-
7
Essence of Number
9
9
9

 

 

T
=
2
-
3
THE
33
15
6
S
=
1
-
5
SPEED
49
13
4
O
=
6
-
2
OF
21
12
3
L
=
3
-
5
LIGHT
56
29
2
-
-
12
-
15
First Total
159
69
15
-
-
1+2
-
1+5
Add to Reduce
1+5+9
6+9
1+5
-
-
3
-
6
Second Total
15
15
6
-
-
-
-
-
Reduce to Deduce
1+5
1+5
-
-
-
3
-
6
Essence of Number
6
6
6

 

 

S
=
1
-
5
SPEED
49
22
4
O
=
6
-
2
OF
21
12
3
L
=
3
-
5
LIGHT
56
29
2
-
-
10
-
12
-
126
63
9
-
-
1+0
-
1+2
-
1+2+6
6+3
-
-
-
1
-
3
-
9
9
9

 

 

-T
12
S
P
E
E
D
-
O
F
-
L
I
G
H
T
-
-
-
-
-
--
-
-
-
-
-
`-
1
-
-
-
-
-
6
-
-
-
9
-
8
-
+
=
24
2+4
=
6
-
6
-
6
-
`-
19
-
-
-
-
-
15
-
-
-
9
-
8
-
+
=
51
5+1
=
6
-
6
-
6
-
12
S
P
E
E
D
-
O
F
-
L
I
G
H
T
-
-
-
-
-
--
-
-
-
-
-
-
-
7
5
5
4
-
-
6
-
3
-
7
-
2
+
=
39
3+9
=
12
1+2
3
-
3
-
`-
-
16
5
5
4
-
-
6
-
12
-
7
-
20
+
=
75
7+5
=
12
1+2
3
-
3
-
12
S
P
E
E
D
-
O
F
-
L
I
G
H
T
-T
-
-
-
-
-
-
-
-
-
-
`-
19
16
5
5
4
-
15
6
-
12
9
7
8
20
+
=
126
1+2+6
=
9
-
9
-
9
-
-
1
7
5
5
4
-T
6
6
-T
3
9
7
8
2
+
=
63
6+3
=
9
-
9
-
9
-
12
S
P
E
E
D
-
O
F
-
L
I
G
H
T
-T
-
-
-
-
-
-
-
-
-
1
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
2
occurs
x
1
=
2
=
2
3
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
4
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
5
-
-
-
5
5
-
-
-
-
-
-
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
6
-
-
-
-
-
-
-
6
6
-
-
-
-
-
-
-
-
6
occurs
x
2
=
12
1+2
3
7
--
-
7
-
-
-
-
-
-
-
-
-
7
-
-
-
-
7
occurs
x
2
=
14
1+4
5
8
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
8
occurs
x
1
=
8
=
8
9
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
9
occurs
x
1
=
27
=
9
45
12
S
P
E
E
D
-
O
F
-
L
I
G
H
T
-
-
45
-
1
12
-
81
-
36
4+5
1+2
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
4+5
-
-
1+2
-
8+1
-
3+4
9
3
S
P
E
E
D
-
O
F
-
L
I
G
H
T
-
-
9
-
1
3
-
9
-
9
-
-
1
7
5
5
4
-
6
6
-
3
9
7
8
2
-
-
-
-
-
-
-
-
-
-
9
3
S
P
E
E
D
-
O
F
-
L
I
G
H
T
-
-
9
-
-
3
-
9
-
9

 

 

 

P
=
7
-
7
PROTONS
117
36
9
A
=
1
-
2
AT
21
12
3
T
=
2
-
3
THE
33
15
6
S
=
1
-
5
SPEED
49
13
4
O
=
6
-
2
OF
21
12
3
L
=
3
-
5
LIGHT
56
29
2
-
-
20
-
24
First Total
297
117
27
-
-
2+0
-
2+4
Add to Reduce
2+9+7
1+1+7
2+7
-
-
2
-
6
Second Total
18
9
9
-
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
2
-
6
Essence of Number
9
9
9

 

 

P
=
7
-
7
PROTONS
117
36
9
F
=
6
-
5
FIRED
42
33
6
A
=
1
-
2
AT
21
12
3
T
=
2
-
3
THE
33
15
6
S
=
1
-
5
SPEED
49
13
4
O
=
6
-
2
OF
21
12
3
L
=
3
-
5
LIGHT
56
29
2
-
-
26
-
29
First Total
339
150
33
-
-
2+6
-
2+9
Add to Reduce
3+3+9
1+5+0
3+3
-
-
8
-
11
Second Total
15
6
6
-
-
-
-
1+1
Reduce to Deduce
1+5
-
-
-
-
8
-
2
Essence of Number
6
6
6

 

 

3
THE
33
15
6
3
LAW
36
9
9
2
OF
21
12
3
6
KARMAS
63
18
9
14
Add to Reduce
153
54
27
1+4
Reduce to Deduce
1+5+3
5+4
2+7
5
Essence of Number
9
9
9

 

S
=
1
-
5
SPEED
49
13
4
O
=
6
-
2
OF
21
12
3
L
=
3
-
5
LIGHT
56
29
2
-
-
10
-
12
Add to Reduce
126
54
9
-
-
1+0
-
1+2
Reduce to Deduce
1+2+6
5+4
-
-
-
1
-
3
Essence of Number
9
9
6

 

 

H
=
8
-
5
HIGGS
50
32
5
B
=
2
-
5
BOSON
65
20
2
P
=
7
-
8
PARTICLE
84
39
3
-
-
17
-
18
First Total
199
91
10
-
-
1+7
-
1+8
Add to Reduce
1+9+9
9+1
1+0
-
-
8
-
9
Second Total
10
10
1
-
-
-
-
-
Reduce to Deduce
1+0
1+0
-
-
-
8
-
9
Essence of Number
1
1
1

 

 

-T
18
H
I
G
G
S
-
B
O
S
O
N
-
P
A
R
T
I
C
L
E
-
-
-
-
-
-
-
-
-
-T
-
8
9
-
-
1
-
-
6
1
6
5
-
-
-
-
-
9
-
-
-
+
=
45
4+5
=
9
=
9
=
9
-
-
8
9
-
-
19
-
-
15
19
15
14
-
-
-
-
-
9
-
-
-
+
=
108
1+0+8
=
9
=
9
=
9
-
18
H
I
G
G
S
-
B
O
S
O
N
-
P
A
R
T
I
C
L
E
-
-
-
-
-
---
-
-
-
-
-
-
-
-
7
7
-
-
2
-
-
-
-
-
7
1
9
2
-
3
3
5
+
=
46
4+6
=
10
1+0
1
=
1
-
-
-
-
7
7
-
-
2
-
-
-
-
-
16
1
18
20
-
3
12
5
+
=
91
9+1
=
10
1+0
1
=
1
-
18
H
I
G
G
S
-
B
O
S
O
N
-
P
A
R
T
I
C
L
E
-
-
-
-
-
-
-
-
-
-
-
-
8
9
7
7
19
-
2
15
19
15
14
-
16
1
18
20
9
3
12
5
+
=
199
1+9+9
=
19
1+9
10
1+0
1
-
-
8
9
7
7
1
-T
2
6
1
6
5
-T
7
1
9
2
9
3
3
5
+
=
91
9+1
=
10
1+0
1
=
1
-
18
H
I
G
G
S
-
B
O
S
O
N
-
P
A
R
T
I
C
L
E
-T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
3
=
3
=
3
-
``-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
3
-
-
-
3
occurs
x
2
=
6
=
6
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
-
-
-
-
6
-
6
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
2
=
12
1+2
3
-
-
-
-
7
7
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
7
occurs
x
3
=
21
2+1
3
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
9
-
9
-
-
-
-
-
9
occurs
x
3
=
27
=
9
4
18
H
I
G
G
S
-
B
O
S
O
N
-
P
A
R
T
I
C
L
E
-
-
41
-
1
18
-
91
-
37
-
1+8
-
9
-
-
-
-
-
-
-
-
-
-
-
-
9
-
9
-
-
-
-
-
4+1
-
-
1+8
-
9+1
-
3+7
4
9
H
I
G
G
S
-
B
O
S
O
N
-
P
A
R
T
I
C
L
E
-
-
5
-
1
9
-
10
-
10
-
-
8
9
7
7
1
-
2
6
1
6
5
-
7
1
9
2
9
3
-
5
-
-
-
-
-
-
-
1+0
-
1+0
4
9
H
I
G
G
S
-
B
O
S
O
N
-
P
A
R
T
I
C
L
E
-
-
5
-
-
9
-
1
-
1

 

 

18
H
I
G
G
S
-
B
O
S
O
N
-
P
A
R
T
I
C
L
E
-
-
-
-
-
-
-
-
-
-
8
9
-
-
1
-
-
6
1
6
5
-
-
-
-
-
9
-
-
-
+
=
45
4+5
=
9
=
9
=
9
-
8
9
-
-
19
-
-
15
19
15
14
-
-
-
-
-
9
-
-
-
+
=
108
1+0+8
=
9
=
9
=
9
18
H
I
G
G
S
-
B
O
S
O
N
-
P
A
R
T
I
C
L
E
-
-
-
-
-
-
-
-
-
-
-
-
-
7
7
-
-
2
-
-
-
-
-
7
1
9
2
-
3
3
5
+
=
46
4+6
=
10
1+0
1
=
1
-
-
-
7
7
-
-
2
-
-
-
-
-
16
1
18
20
-
3
12
5
+
=
91
9+1
=
10
1+0
1
=
1
18
H
I
G
G
S
-
B
O
S
O
N
-
P
A
R
T
I
C
L
E
-
-
-
-
-
-
-
-
-
-
-
8
9
7
7
19
-
2
15
19
15
14
-
16
1
18
20
9
3
12
5
+
=
199
1+9+9
=
19
1+9
10
1+0
1
-
8
9
7
7
1
-T
2
6
1
6
5
-T
7
1
9
2
9
3
3
5
+
=
91
9+1
=
10
1+0
1
=
1
18
H
I
G
G
S
-
B
O
S
O
N
-
P
A
R
T
I
C
L
E
-T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
-
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
3
=
3
=
3
``-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
3
-
-
-
3
occurs
x
2
=
6
=
6
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
-
-
-
6
-
6
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
2
=
12
1+2
3
-
-
-
7
7
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
7
occurs
x
3
=
21
2+1
3
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
9
-
9
-
-
-
-
-
9
occurs
x
3
=
27
=
9
18
H
I
G
G
S
-
B
O
S
O
N
-
P
A
R
T
I
C
L
E
-
-
41
-
1
18
-
91
-
37
1+8
-
9
-
-
-
-
-
-
-
-
-
-
-
-
9
-
9
-
-
-
-
-
4+1
-
-
1+8
-
9+1
-
3+7
9
H
I
G
G
S
-
B
O
S
O
N
-
P
A
R
T
I
C
L
E
-
-
5
-
1
9
-
10
-
10
-
8
9
7
7
1
-
2
6
1
6
5
-
7
1
9
2
9
3
-
5
-
-
-
-
-
-
-
1+0
-
1+0
9
H
I
G
G
S
-
B
O
S
O
N
-
P
A
R
T
I
C
L
E
-
-
5
-
-
9
-
1
-
1

 

 

18
H
I
G
G
S
B
O
S
O
N
P
A
R
T
I
C
L
E
-
-
-
-
-
-
-
-
-
-
8
9
-
-
1
-
6
1
6
5
-
-
-
-
9
-
-
-
+
=
45
4+5
=
9
=
9
=
9
-
8
9
-
-
19
-
15
19
15
14
-
-
-
-
9
-
-
-
+
=
108
1+0+8
=
9
=
9
=
9
18
H
I
G
G
S
B
O
S
O
N
P
A
R
T
I
C
L
E
-
-
-
-
-
--
-
-
-
-
-
-
-
7
7
-
2
-
-
-
-
7
1
9
2
-
3
3
5
+
=
46
4+6
=
10
1+0
1
=
1
-
-
-
7
7
-
2
-
-
-
-
16
1
18
20
-
3
12
5
+
=
91
9+1
=
10
1+0
1
=
1
18
H
I
G
G
S
B
O
S
O
N
P
A
R
T
I
C
L
E
-
-
-
-
-
-
-
-
-
-
-
8
9
7
7
19
2
15
19
15
14
16
1
18
20
9
3
12
5
+
=
199
1+9+9
=
19
1+9
10
1+0
1
-
8
9
7
7
1
2
6
1
6
5
7
1
9
2
9
3
3
5
+
=
91
9+1
=
10
1+0
1
=
1
18
H
I
G
G
S
B
O
S
O
N
P
A
R
T
I
C
L
E
-T
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
-
1
-
-
-
-
-
-
-
-
1
occurs
x
3
=
3
=
3
``-
-
-
-
-
-
2
-
-
-
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
3
-
-
-
3
occurs
x
2
=
6
=
6
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
-
-
-
5
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
-
-
6
-
6
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
2
=
12
1+2
3
-
-
-
7
7
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
7
occurs
x
3
=
21
2+1
3
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
9
-
-
-
-
-
-
-
-
-
-
9
-
9
-
-
-
-
-
9
occurs
x
3
=
27
=
9
18
H
I
G
G
S
B
O
S
O
N
P
A
R
T
I
C
L
E
-
-
41
-
1
18
-
91
-
37
1+8
-
9
-
-
-
-
-
-
-
-
-
-
9
-
9
-
-
-
-
-
4+1
-
-
1+8
-
9+1
-
3+7
9
H
I
G
G
S
B
O
S
O
N
P
A
R
T
I
C
L
E
-
-
5
-
1
9
-
10
-
10
-
8
9
7
7
1
2
6
1
6
5
7
1
9
2
9
3
-
5
-
-
-
-
-
-
-
1+0
-
1+0
9
H
I
G
G
S
B
O
S
O
N
P
A
R
T
I
C
L
E
-
-
5
-
-
9
-
1
-
1

 

 

T
=
2
-
3
THE
33
15
6
G
=
7
-
3
GOD
26
26
8
P
=
7
-
8
PARTICLE
84
39
3
-
-
16
-
14
Add to Reduce
143
80
17
-
-
1+6
-
1+4
Reduce to Deduce
1+4+3
8+0
1+7
-
-
7
-
5
Essence of Number
8
8
8

 

 

T
=
2
-
3
THE
33
15
6
G
=
7
-
3
GOD
26
26
8
P
=
7
-
9
PARTICLES
103
40
4
-
-
16
-
15
Add to Reduce
162
81
18
-
-
1+6
-
1+5
Reduce to Deduce
1+6+2
8+1
1+8
-
-
7
-
6
Essence of Number
9
9
9

 

 

Higgs boson - Wikipedia
https://en.wikipedia.org/wiki/Higgs_boson

The Higgs boson is an elementary particle in the Standard Model of particle physics. First suspected to exist in the 1960s, it is the quantum excitation of the Higgs field, a fundamental field of crucial importance to particle physics theory. Unlike other known fields such as the electromagnetic field, it has a non-zero constant ...

Higgs boson
Subatomic particle

The Higgs boson is an elementary particle in the Standard Model of particle physics. First suspected to exist in the 1960s, it is the quantum excitation of the Higgs field, a fundamental field of crucial importance to particle physics theory. Wikipedia

Composition: Elementary particle

Classification: Boson

Symbol: H°

Mass: 125.09±0.21 (stat.)±0.11 (syst.) GeV/c² (CMS+ATLAS)

Electric charge: 0 e

Discovered: Large Hadron Collider (2011–2013)

Mean lifetime: 1.56×10-22 s (predicted)

 

T
=
2
-
3
THE
33
15
6
H
=
8
-
5
HIGGS
50
41
5
B
=
2
-
5
BOSON
65
29
2
-
-
12
-
13
First Total
148
85
13
-
-
1+2
-
1+3
Add to Reduce
1+4+8
8+5
1+3
-
-
3
-
4
Second Total
13
13
4
-
-
-
-
-
Reduce to Deduce
1+3
1+3
-
-
-
3
-
4
Essence of Number
4
4
4

 

 

-
-
-
-
-
THE HIGGS BOSON
-
-
-
-
1
2
3
4
5
6
7
8
9
-
-
-
THE
-
-
-
-
-
-
3
4
-
-
-
-
-
T
=
6
1
1
T
20
2
2
-
-
2
3
4
-
-
-
-
-
H
=
8
2
1
H
8
8
8
-
-
-
3
4
-
-
-
8
-
E
=
5
3
1
E
5
5
5
-
-
-
3
4
5
-
-
-
-
15
-
3
THE
33
15
15
-
-
-
3
4
-
-
-
-
-
-
-
-
-
-
HIGGS
-
-
-
-
-
-
3
4
-
-
-
-
-
H
=
8
4
1
H
8
8
8
-
-
-
3
4
-
-
-
8
-
I
=
9
5
1
I
9
9
9
-
-
-
3
4
-
-
-
-
9
G
=
7
6
1
G
7
7
7
-
-
-
3
4
-
-
7
-
-
G
=
7
7
1
G
7
7
7
-
-
-
3
4
-
-
7
-
-
S
=
1
8
1
S
19
10
1
-
1
-
3
4
-
-
-
-
-
-
-
32
-
5
HIGGS
50
41
32
-
-
-
3
4
-
-
-
-
-
-
-
-
-
-
BOSON
-
-
-
-
-
-
3
4
-
-
-
-
-
B
=
2
9
1
B
2
2
2
-
-
2
3
4
-
-
-
-
-
O
=
6
10
1
O
15
6
6
-
-
-
3
4
-
6
-
-
-
S
=
1
11
1
S
19
10
1
-
1
-
3
4
-
-
-
-
-
O
=
6
12
1
O
15
6
6
-
-
-
3
4
-
6
-
-
-
N
=
5
13
1
N
14
5
5
-
-
-
3
4
5
-
-
-
-
20
-
5
BOSON
65
29
20
-
-
-
3
4
-
-
-
-
-
-
-
67
-
13
THE HIGGS BOSON
148
85
67
-
2
4
3
4
10
12
14
16
9
6+7
-
1+3
-
1+4+8
8+5
6+7
-
-
-
-
-
1+0
1+2
1+4
1+6
-
-
-
13
-
4
THE HIGGS BOSON
13
13
13
-
2
4
3
4
1
3
5
7
9
-
-
1+3
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
4
THE HIGGS BOSON
4
4
4
-
2
4
3
4
1
3
5
7
9

 

 

-
-
-
-
-
THE HIGGS BOSON
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
6
1
1
T
20
2
2
-
-
2
3
4
-
-
-
-
-
H
=
8
2
1
H
8
8
8
-
-
-
3
4
-
-
-
8
-
E
=
5
3
1
E
5
5
5
-
-
-
3
4
5
-
-
-
-
H
=
8
4
1
H
8
8
8
-
-
-
3
4
-
-
-
8
-
I
=
9
5
1
I
9
9
9
-
-
-
3
4
-
-
-
-
9
G
=
7
6
1
G
7
7
7
-
-
-
3
4
-
-
7
-
-
G
=
7
7
1
G
7
7
7
-
-
-
3
4
-
-
7
-
-
S
=
1
8
1
S
19
10
1
-
1
-
3
4
-
-
-
-
-
B
=
2
9
1
B
2
2
2
-
-
2
3
4
-
-
-
-
-
O
=
6
10
1
O
15
6
6
-
-
-
3
4
-
6
-
-
-
S
=
1
11
1
S
19
10
1
-
1
-
3
4
-
-
-
-
-
O
=
6
12
1
O
15
6
6
-
-
-
3
4
-
6
-
-
-
N
=
5
13
1
N
14
5
5
-
-
-
3
4
5
-
-
-
-
-
-
67
-
13
THE HIGGS BOSON
148
85
67
-
2
4
3
4
10
12
14
16
9
6+7
-
1+3
-
1+4+8
8+5
6+7
-
-
-
-
-
1+0
1+2
1+4
1+6
-
-
-
13
-
4
THE HIGGS BOSON
13
13
13
-
2
4
3
4
1
3
5
7
9
-
-
1+3
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
4
THE HIGGS BOSON
4
4
4
-
2
4
3
4
1
3
5
7
9

 

 

-
-
-
-
-
THE HIGGS BOSON
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
8
1
S
19
10
1
-
1
-
3
4
-
-
-
-
-
S
=
1
11
1
S
19
10
1
-
1
-
3
4
-
-
-
-
-
T
=
6
1
1
T
20
2
2
-
-
2
3
4
-
-
-
-
-
B
=
2
9
1
B
2
2
2
-
-
2
3
4
-
-
-
-
-
E
=
5
3
1
E
5
5
5
-
-
-
3
4
5
-
-
-
-
N
=
5
13
1
N
14
5
5
-
-
-
3
4
5
-
-
-
-
O
=
6
10
1
O
15
6
6
-
-
-
3
4
-
6
-
-
-
O
=
6
12
1
O
15
6
6
-
-
-
3
4
-
6
-
-
-
G
=
7
6
1
G
7
7
7
-
-
-
3
4
-
-
7
-
-
G
=
7
7
1
G
7
7
7
-
-
-
3
4
-
-
7
-
-
H
=
8
2
1
H
8
8
8
-
-
-
3
4
-
-
-
8
-
H
=
8
4
1
H
8
8
8
-
-
-
3
4
-
-
-
8
-
I
=
9
5
1
I
9
9
9
-
-
-
3
4
-
-
-
-
9
-
-
67
-
13
THE HIGGS BOSON
148
85
67
-
2
4
3
4
10
12
14
16
9
6+7
-
1+3
-
1+4+8
8+5
6+7
-
-
-
-
-
1+0
1+2
1+4
1+6
-
-
-
13
-
4
THE HIGGS BOSON
13
13
13
-
2
4
3
4
1
3
5
7
9
-
-
1+3
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
4
THE HIGGS BOSON
4
4
4
-
2
4
3
4
1
3
5
7
9

 

LETTERS RE-ARRANGED IN NUMERICAL ORDER

 

-
-
-
-
-
THE HIGGS BOSON
-
-
-
-
1
2
5
6
7
8
9
S
=
1
8
1
S
19
10
1
-
1
-
-
-
-
-
-
S
=
1
11
1
S
19
10
1
-
1
-
-
-
-
-
-
T
=
6
1
1
T
20
2
2
-
-
2
-
-
-
-
-
B
=
2
9
1
B
2
2
2
-
-
2
-
-
-
-
-
E
=
5
3
1
E
5
5
5
-
-
-
5
-
-
-
-
N
=
5
13
1
N
14
5
5
-
-
-
5
-
-
-
-
O
=
6
10
1
O
15
6
6
-
-
-
-
6
-
-
-
O
=
6
12
1
O
15
6
6
-
-
-
-
6
-
-
-
G
=
7
6
1
G
7
7
7
-
-
-
-
-
7
-
-
G
=
7
7
1
G
7
7
7
-
-
-
-
-
7
-
-
H
=
8
2
1
H
8
8
8
-
-
-
-
-
-
8
-
H
=
8
4
1
H
8
8
8
-
-
-
-
-
-
8
-
I
=
9
5
1
I
9
9
9
-
-
-
-
-
-
-
9
-
-
67
-
13
THE HIGGS BOSON
148
85
67
-
2
4
10
12
14
16
9
6+7
-
1+3
-
1+4+8
8+5
6+7
-
-
-
1+0
1+2
1+4
1+6
-
-
-
13
-
4
THE HIGGS BOSON
13
13
13
-
2
4
1
3
5
7
9
-
-
1+3
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
4
-
4
THE HIGGS BOSON
4
4
4
-
2
4
1
3
5
7
9

 

 

H
=
8
-
5
HIGGS
50
32
5
B
=
2
-
5
BOSON
65
20
2
P
=
7
-
8
PARTICLE
84
39
3
-
-
17
-
18
First Total
199
91
10
-
-
1+7
-
1+8
Add to Reduce
1+9+9
9+1
1+0
-
-
8
-
18
Second Total
10
10
1
-
-
-
-
1+8
Reduce to Deduce
1+0
1+0
-
-
-
8
-
9
Essence of Number
1
1
1

 

 

-
8
H
Y
D
R
O
G
E
N
-
-
-
-
-
--
-
-
-
-
-
-
8
-
-
-
6
-
-
5
+
=
19
1+9
=
10
1+0
1
=
1
-
-
8
-
-
-
15
-
-
14
+
=
37
3+7
=
10
1+0
1
=
1
-
8
H
Y
D
R
O
G
E
N
-
-
-
-
-
--
-
-
-
-
-
-
-
7
4
9
-
7
5
-
+
=
32
3+2
=
5
-
5
=
5
-
-
-
25
4
18
-
7
5
-
+
=
59
5+9
=
14
1+4
5
=
5
-
8
H
Y
D
R
O
G
E
N
-
-
-
-
-
--
-
-
-
-
-
-
8
25
4
18
15
7
5
14
+
=
96
9+6
=
15
1+5
6
=
6
-
-
8
7
3
9
6
7
5
5
+
=
51
5+1
=
6
-
6
=
6
-
8
H
Y
D
R
O
G
E
N
-
-
-
-
-
--
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
1
ONE
1
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
2
TWO
2
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
3
THREE
3
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
-
5
5
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
-
6
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
-
7
-
-
-
7
-
-
-
-
7
occurs
x
2
=
14
1+4
5
-
-
8
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
8
-
-
-
-
-
9
-
-
-
-
-
-
9
occurs
x
1
=
9
-
9
6
8
H
Y
D
R
O
G
E
N
-
-
39
-
-
8
-
51
-
33
-
-
-
-
-
9
-
-
-
-
-
-
3+9
-
-
-
-
5+1
-
3+3
6
8
H
Y
D
R
O
G
E
N
-
-
12
-
-
8
-
6
-
6
-
-
8
7
3
9
6
7
5
5
-
-
1+2
-
-
-
-
-
-
-
6
8
H
Y
D
R
O
G
E
N
-
-
3
-
-
8
-
6
-
6

 

 

8
H
Y
D
R
O
G
E
N
-
-
-
-
-
--
-
-
-
-
-
8
-
-
-
6
-
-
5
+
=
19
1+9
=
10
1+0
1
=
1
-
8
-
-
-
15
-
-
14
+
=
37
3+7
=
10
1+0
1
=
1
8
H
Y
D
R
O
G
E
N
-
-
-
-
-
--
-
-
-
-
-
-
7
4
9
-
7
5
-
+
=
32
3+2
=
5
-
5
=
5
-
-
25
4
18
-
7
5
-
+
=
59
5+9
=
14
1+4
5
=
5
8
H
Y
D
R
O
G
E
N
-
-
-
-
-
--
-
-
-
-
-
8
25
4
18
15
7
5
14
+
=
96
9+6
=
15
1+5
6
=
6
-
8
7
3
9
6
7
5
5
+
=
51
5+1
=
6
-
6
=
6
8
H
Y
D
R
O
G
E
N
-
-
-
-
-
--
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
4
occurs
x
1
=
4
=
4
-
-
-
-
-
-
-
5
5
-
-
5
occurs
x
2
=
10
1+0
1
-
-
-
-
-
6
-
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
7
-
-
-
7
-
-
-
-
7
occurs
x
2
=
14
1+4
5
-
8
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
8
-
-
-
-
9
-
-
-
-
-
-
9
occurs
x
1
=
9
-
9
8
H
Y
D
R
O
G
E
N
-
-
39
-
-
8
-
51
-
33
-
-
-
-
9
-
-
-
-
-
-
3+9
-
-
-
-
5+1
-
3+3
8
H
Y
D
R
O
G
E
N
-
-
12
-
-
8
-
6
-
6
-
8
7
3
9
6
7
5
5
-
-
1+2
-
-
-
-
-
-
-
8
H
Y
D
R
O
G
E
N
-
-
3
-
-
8
-
6
-
6

 

The hydrogen line, 21-centimeter line, is the electromagnetic radiation spectral line

 

-
-
-
-
-
-
-
-
-
T
=
2
-
3
THE
33
15
6
H
=
8
-
8
HYDROGEN
96
51
6
L
=
3
-
4
LINE
40
22
4
T
=
2
-
9
TWENTYONE
141
42
6
C
=
3
-
10
CENTIMETER
112
49
4
L
=
3
-
4
LINE
40
22
4
I
=
9
-
2
IS
28
10
1
T
=
2
-
3
THE
33
15
6
E
=
5
-
15
ELECTROMAGNETIC
150
69
6
R
=
9
-
9
RADIATION
91
46
1
S
=
1
-
8
SPECTRAL
94
31
4
L
=
3
-
4
LINE
40
22
4
-
-
50
-
79
First Total
898
394
52
-
-
5+0
-
7+9
Add to Reduce
8+9+9
3+9+4
5+2
-
-
5
-
16
Second Total
25
16
7
-
-
-
-
1+6
Reduce to Deduce
2+5
1+6
-
-
-
5
-
7
Essence of Number
7
7
7

 

 

Hydrogen line - Wikipedia

https://en.wikipedia.org › wiki › Hydrogen_line
The hydrogen line, 21-centimeter line, or H I line is the electromagnetic radiation spectral line ... This wavelength falls within the microwave region of the electromagnetic spectrum, and it is ... Despite these problems, 21 cm observations, along with space-based gravitational wave observations, are generally viewed as the ...

 

21-centimeter line, or H I line is the electromagnetic radiation spectral line

 

-
-
-
-
-
-
-
-
-
T
=
2
-
9
TWENTYONE
141
42
6
C
=
3
-
10
CENTIMETER
112
49
4
L
=
3
-
4
LINE
40
22
4
-
-
8
-
23
First Total
293
113
14
-
-
-
-
2+3
Add to Reduce
2+9+3
1+1+3
1+4
-
-
8
-
5
Second Total
14
5
5
-
-
-
-
-
Reduce to Deduce
1+4
-
-
-
-
8
-
5
Essence of Number
5
5
5

 

The hydrogen line, 21-centimeter line, or H I line[1] is the electromagnetic radiation spectral line that is created by a change in the energy state of neutral hydrogen atoms. This electromagnetic radiation is at the precise frequency of 1,420,405,751.7667±0.0009 Hz,[2][3] which is equivalent to the vacuum wavelength of 21.1061140542 cm in free space. This wavelength falls within the microwave region of the electromagnetic spectrum, and it is observed frequently in radio astronomy because those radio waves can penetrate the large clouds of interstellar cosmic dust that are opaque to visible light. This line is also the theoretical basis of hydrogen maser.

The microwaves of the hydrogen line come from the atomic transition of an electron between the two hyperfine levels of the hydrogen 1 s ground state that have an energy difference of ˜ 5.87433 µeV [9.411708×10-25 J].[4] It is called the spin-flip transition. The frequency, ?, of the quanta that are emitted by this transition between two different energy levels is given by the Planck–Einstein relation E = h?. According to that relation, the photon energy of a 1,420,405,751.7667 Hz photon is ˜ 5.87433 µeV [9.411708×10-25 J]. The constant of proportionality, h, is known as the Planck constant.
Further information: Hyperfine structure

The ground state of neutral hydrogen consists of an electron bound to a proton. Both the electron and the proton have intrinsic magnetic dipole moments ascribed to their spin, whose interaction results in a slight increase in energy when the spins are parallel, and a decrease when antiparallel. The fact that only parallel and antiparallel states are allowed is a result of the quantum mechanical discretization of the total angular momentum of the system. When the spins are parallel, the magnetic dipole moments are antiparallel (because the electron and proton have opposite charge), thus one would expect this configuration to actually have lower energy just as two magnets will align so that the north pole of one is closest to the south pole of the other. This logic fails here because the wave functions of the electron and the proton overlap; that is, the electron is not spatially displaced from the proton, but encompasses it. The magnetic dipole moments are therefore best thought of as tiny current loops. As parallel currents attract, the parallel magnetic dipole moments (i.e., antiparallel spins) have lower energy.[5] The transition has an energy difference of 5.87433 µeV that when applied in the Planck equation gives:
? = 1 ? · c = h E · c = 4.1357 · 10 - 15 eV s 5.874 33 · 10 - 6 eV · 2.9979 · 10 8 m s - 1 = 0.211 06 m = 21.106 cm {\displaystyle \lambda ={\frac {1}{\nu }}\cdot c={\frac {h}{E}}\cdot c={\frac {4.1357\cdot 10^{-15}\;{\text{eV}}\;{\text{s}}}{5.874\,33\cdot 10^{-6}\;{\text{eV}}}}\cdot 2.9979\cdot 10^{8}\;{\text{m}}\;{\text{s}}^{-1}=0.211\,06\;{\text{m}}=21.106\;{\text{cm}}} {\displaystyle \lambda ={\frac {1}{\nu }}\cdot c={\frac {h}{E}}\cdot c={\frac {4.1357\cdot 10^{-15}\;{\text{eV}}\;{\text{s}}}{5.874\,33\cdot 10^{-6}\;{\text{eV}}}}\cdot 2.9979\cdot 10^{8}\;{\text{m}}\;{\text{s}}^{-1}=0.211\,06\;{\text{m}}=21.106\;{\text{cm}}}
where h is the Planck constant and c is the speed of light.

This transition is highly forbidden with an extremely small transition rate of 2.9×10-15 s-1,[6] and a mean lifetime of the excited state of around 10 million years. A spontaneous occurrence of the transition is unlikely to be seen in a laboratory on Earth, but it can be artificially induced using a hydrogen maser. It is commonly observed in astronomical settings such as hydrogen clouds in our galaxy and others. Owing to its long lifetime, the line has an extremely small natural width, so most broadening is due to Doppler shifts caused by bulk motion or nonzero temperature of the emitting regions.

Discovery?[edit]

During the 1930s, it was noticed that there was a radio 'hiss' that varied on a daily cycle and appeared to be extraterrestrial in origin. After initial suggestions that this was due to the Sun, it was observed that the radio waves seemed to propagate from the centre of the Galaxy. These discoveries were published in 1940 and were noted by Jan Oort who knew that significant advances could be made in astronomy if there were emission lines in the radio part of the spectrum. He referred this to Hendrik van de Hulst who, in 1944, predicted that neutral hydrogen could produce radiation at a frequency of 1420.4058 MHz due to two closely spaced energy levels in the ground state of the hydrogen atom.

The 21 cm line (1420.4 MHz) was first detected in 1951 by Ewen and Purcell at Harvard University,[7] and published after their data was corroborated by Dutch astronomers Muller and Oort,[8] and by Christiansen and Hindman in Australia. After 1952 the first maps of the neutral hydrogen in the Galaxy were made, and revealed for the first time the spiral structure of the Milky Way.
Uses?[edit]

In radio astronomy?[edit]

The 21 cm spectral line appears within the radio spectrum (in the L band of the UHF band of the microwave window to be exact). Electromagnetic energy in this range can easily pass through the Earth's atmosphere and be observed from the Earth with little interference.

Assuming that the hydrogen atoms are uniformly distributed throughout the galaxy, each line of sight through the galaxy will reveal a hydrogen line. The only difference between each of these lines is the doppler shift that each of these lines has. Hence, one can calculate the relative speed of each arm of our galaxy. The rotation curve of our galaxy has been calculated using the 21 cm hydrogen line. It is then possible to use the plot of the rotation curve and the velocity to determine the distance to a certain point within the galaxy.

Hydrogen line observations have also been used indirectly to calculate the mass of galaxies, to put limits on any changes over time of the universal gravitational constant and to study the dynamics of individual galaxies.

In cosmology?[edit]

The line is of great interest in Big Bang cosmology because it is the only known way to probe the "dark ages" from recombination to reionization. Including the redshift, this line will be observed at frequencies from 200 MHz to about 9 MHz on Earth. It potentially has two applications. First, by mapping the intensity of redshifted 21 centimeter radiation it can, in principle, provide a very precise picture of the matter power spectrum in the period after recombination. Second, it can provide a picture of how the universe was reionized, as neutral hydrogen which has been ionized by radiation from stars or quasars will appear as holes in the 21 cm background.

However, 21 cm observations are very difficult to make. Ground-based experiments to observe the faint signal are plagued by interference from television transmitters and the ionosphere, so they must be made from very secluded sites with care taken to eliminate interference. Space based experiments, even on the far side of the Moon (where they would be sheltered from interference from terrestrial radio signals), have been proposed to compensate for this. Little is known about other effects, such as synchrotron emission and free–free emission on the galaxy. Despite these problems, 21 cm observations, along with space-based gravitational wave observations, are generally viewed as the next great frontier in observational cosmology, after the cosmic microwave background polarization.

Relevance to the search for non-human intelligent life?[edit]


The hyperfine transition of hydrogen, as depicted on the Pioneer and Voyager spacecraft.
The Pioneer plaque, attached to the Pioneer 10 and Pioneer 11 spacecraft, portrays the hyperfine transition of neutral hydrogen and used the wavelength as a standard scale of measurement. For example, the height of the woman in the image is displayed as eight times 21 cm, or 168 cm. Similarly the frequency of the hydrogen spin-flip transition was used for a unit of time in a map to Earth included on the Pioneer plaques and also the Voyager 1 and Voyager 2 probes. On this map, the position of the Sun is portrayed relative to 14 pulsars whose rotation period circa 1977 is given as a multiple of the frequency of the hydrogen spin-flip transition. It is theorized by the plaque's creators that an advanced civilization would then be able to use the locations of these pulsars to locate the Solar System at the time the spacecraft were launched.

The 21 cm hydrogen line is considered a favorable frequency by the SETI program in their search for signals from potential extraterrestrial civilizations. In 1959, Italian physicist Giuseppe Cocconi and American physicist Philip Morrison published "Searching for Interstellar Communications", a paper proposing the 21 cm hydrogen line and the potential of microwaves in the search for interstellar communications. According to George Basalla, the paper by Cocconi and Morrison "provided a reasonable theoretical basis" for the then-nascent SETI program.[9] Similarly, Pyotr Makovetsky proposed SETI use a frequency which is equal to either
p × 1420.40575177 MHz = 4.46233627 GHz
or
2p × 1420.40575177 MHz = 8.92467255 GHz
Since p is an irrational number, such a frequency could not possibly be produced in a natural way as a harmonic, and would clearly signify its artificial origin. Such a signal would not be overwhelmed by the H I line itself, or by any of its harmonics.[10]

A suggested modification to the mentioned idea is to search on currently unused Sky or other satellite LNB frequencies, as it is possible that with enough synchronized receivers AI inference techniques could be used. In this case, the target frequency would be in the 13 GHz band (13.37–13.39 GHz), derived from 3pH, but there may be merit in searching on other unused bands within the transition areas not normally used, with the PCB also including an onboard reference clock derived from Droitwich or MSF.[citation needed]

See also?[edit]
Chronology of the universe
Dark Ages Radio Explorer
H-alpha, the visible red spectral line with wavelength of 6562.8 ångstroms
Hydrogen spectral series
Rydberg formula
Balmer series
Schelling point
Timeline of the Big Bang

https://en.wikipedia.org › wiki › Hydrogen_line
The hydrogen line, 21-centimeter line, or H I line is the electromagnetic radiation spectral line ... This wavelength falls within the microwave region of the electromagnetic spectrum, and it is ... Despite these problems, 21 cm observations, along with space-based gravitational wave observations, are generally viewed as the ...

21-cm Line Radiation
In 1944 Hendrik van de Hulst predicted that the cold atomic hydrogen (H I) gas should emit a particular wavelength of radio energy from a slight energy change in the hydrogen atoms. The wavelength is 21.1 centimeters (frequency = 1420.4 MHz) so this radiation is called 21-cm line radiation.28 May 2019

 

T
=
2
-
9
TWENTYONE
141
42
6
C
=
3
-
10
CENTIMETER
112
49
4
L
=
3
-
4
LINE
40
22
4
I
=
9
-
2
IS
28
10
1
T
=
2
-
3
THE
33
15
6
E
=
5
-
15
ELECTROMAGNETIC
150
69
6
R
=
9
-
9
RADIATION
91
46
1
S
=
1
-
8
SPECTRAL
94
31
4
L
=
3
-
4
LINE
40
22
4
-
-
37
-
64
First Total
729
306
36
-
-
3+7
-
6+4
Add to Reduce
7+2+9
3+0+6
3+6
-
-
10
-
10
Second Total
18
9
9
-
-
1+0
-
1+0
Reduce to Deduce
1+8
-
-
-
-
1
-
1
Essence of Number
9
9
9

 

TWENTY ONE CENTIMETER LINE

21-centimeter line, or H I line is the electromagnetic radiation spectral line

 

T
=
2
-
9
TWENTYONE
141
42
6
-
1
2
3
4
5
6
7
8
9
C
=
3
-
10
CENTIMETER
112
49
4
-
-
-
-
-
-
-
-
-
-
L
=
3
-
4
LINE
40
22
4
-
-
-
-
-
-
-
-
-
-
-
-
-5
-
23
First Total
293
113
14
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
T
=
2
1
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
W
=
5
2
1
W
23
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
3
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
4
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
T
=
2
5
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
Y
=
7
6
1
Y
25
7
7
-
-
-
-
-
-
-
7
-
-
O
=
6
7
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
N
=
5
8
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
9
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
42
-
9
-
141
42
42
-
-
-
-
-
-
-
-
-
-
C
=
3
10
1
C
3
3
3
-
-
-
3
-
-
-
-
-
-
E
=
5
11
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
12
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
T
=
2
13
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
I
=
9
14
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
M
=
4
15
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
16
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
T
=
2
17
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
E
=
5
18
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
R
=
9
19
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
-
-
40
-
10
-
112
49
40
-
-
-
-
-
-
-
-
-
-
L
=
3
10
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
I
=
9
14
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
N
=
5
12
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
18
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
22
-
4
-
40
22
22
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
8
6
4
55
6
7
8
27
T
=
2
-
9
TWENTYONE
141
42
6
-
-
-
-
-
5+5
-
-
-
2+7
C
=
3
-
10
CENTIMETER
112
49
4
-
1
8
6
4
10
6
7
8
9
L
=
3
-
4
LINE
40
22
4
-
-
-
-
-
1+0
-
-
-
-
-
-
8
-
23
First Total
293
113
14
-
1
8
6
4
1
6
7
8
9
-
-
-
-
2+3
Add to Reduce
2+9+3
1+1+3
1+4
-
-
-
-
-
-
-
-
-
-
-
-
8
-
5
Second Total
14
5
5
-
1
8
6
4
1
6
7
8
9
-
-
-
-
-
Reduce to Deduce
1+4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
5
Essence of Number
5
5
5
-
1
8
6
4
1
6
7
8
9

 

TWENTY ONE CENTIMETER LINE

 

T
=
2
-
9
TWENTYONE
141
42
6
-
1
2
3
4
5
6
7
8
9
C
=
3
-
10
CENTIMETER
112
49
4
-
-
-
-
-
-
-
-
-
-
L
=
3
-
4
LINE
40
22
4
-
-
-
-
-
-
-
-
-
-
-
-
-5
-
23
First Total
293
113
14
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
T
=
2
1
1
T
20
2
2
-
1
2
-
-
-
-
-
8
-
W
=
5
2
1
W
23
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
3
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
N
=
5
4
1
N
14
5
5
-
1
-
-
-
5
-
-
8
-
T
=
2
5
1
T
20
2
2
-
1
2
-
-
-
-
-
8
-
Y
=
7
6
1
Y
25
7
7
-
1
-
-
-
-
-
7
8
-
O
=
6
7
1
O
15
6
6
-
1
-
-
-
-
6
-
8
-
N
=
5
8
1
N
14
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
9
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
C
=
3
10
1
C
3
3
3
-
1
-
3
-
-
-
-
8
-
E
=
5
11
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
N
=
5
12
1
N
14
5
5
-
1
-
-
-
5
-
-
8
-
T
=
2
13
1
T
20
2
2
-
1
2
-
-
-
-
-
8
-
I
=
9
14
1
I
9
9
9
-
1
-
-
-
-
-
-
8
9
M
=
4
15
1
M
13
4
4
-
1
-
-
4
-
-
-
8
-
E
=
5
16
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
T
=
2
17
1
T
20
2
2
-
1
2
-
-
-
-
-
8
-
E
=
5
18
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
R
=
9
19
1
R
18
9
9
-
1
-
-
-
-
-
-
8
9
L
=
3
10
1
L
12
3
3
-
1
-
3
-
-
-
-
8
-
I
=
9
14
1
I
9
9
9
-
1
-
-
-
-
-
-
8
9
N
=
5
12
1
N
14
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
18
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
-
-
-
-
-
-
-
-
-
-
1
8
6
4
55
6
7
8
27
T
=
2
-
9
TWENTYONE
141
42
6
-
-
-
-
-
5+5
-
-
-
2+7
C
=
3
-
10
CENTIMETER
112
49
4
-
1
8
6
4
10
6
7
8
9
L
=
3
-
4
LINE
40
22
4
-
-
-
-
-
1+0
-
-
-
-
-
-
8
-
23
First Total
293
113
14
-
1
8
6
4
1
6
7
8
9
-
-
-
-
2+3
Add to Reduce
2+9+3
1+1+3
1+4
-
-
-
-
-
-
-
-
-
-
-
-
8
-
5
Second Total
14
5
5
-
1
8
6
4
1
6
7
8
9
-
-
-
-
-
Reduce to Deduce
1+4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
5
Essence of Number
5
5
5
-
1
8
6
4
1
6
7
8
9

 

LETTERS TRANSPOSED INTO NUMBER REARRANGED IN NUMERICAL ORDER

 

T
=
2
-
9
TWENTYONE
141
42
6
-
1
2
3
4
5
6
7
8
9
C
=
3
-
10
CENTIMETER
112
49
4
-
-
-
-
-
-
-
-
-
-
L
=
3
-
4
LINE
40
22
4
-
-
-
-
-
-
-
-
-
-
-
-
-5
-
23
First Total
293
113
14
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
T
=
2
1
1
T
20
2
2
-
1
2
-
-
-
-
-
8
-
T
=
2
5
1
T
20
2
2
-
1
2
-
-
-
-
-
8
-
T
=
2
13
1
T
20
2
2
-
1
2
-
-
-
-
-
8
-
T
=
2
17
1
T
20
2
2
-
1
2
-
-
-
-
-
8
-
C
=
3
10
1
C
3
3
3
-
1
-
3
-
-
-
-
8
-
L
=
3
10
1
L
12
3
3
-
1
-
3
-
-
-
-
8
-
M
=
4
15
1
M
13
4
4
-
1
-
-
4
-
-
-
8
-
W
=
5
2
1
W
23
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
3
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
N
=
5
4
1
N
14
5
5
-
1
-
-
-
5
-
-
8
-
N
=
5
8
1
N
14
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
9
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
11
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
N
=
5
12
1
N
14
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
16
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
18
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
N
=
5
12
1
N
14
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
18
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
O
=
6
7
1
O
15
6
6
-
1
-
-
-
-
6
-
8
-
Y
=
7
6
1
Y
25
7
7
-
1
-
-
-
-
-
7
8
-
I
=
9
14
1
I
9
9
9
-
1
-
-
-
-
-
-
8
9
R
=
9
19
1
R
18
9
9
-
1
-
-
-
-
-
-
8
9
I
=
9
14
1
I
9
9
9
-
1
-
-
-
-
-
-
8
9
-
-
-
-
-
-
-
-
-
-
1
8
6
4
55
6
7
8
27
T
=
2
-
9
TWENTYONE
141
42
6
-
-
-
-
-
5+5
-
-
-
2+7
C
=
3
-
10
CENTIMETER
112
49
4
-
1
8
6
4
10
6
7
8
9
L
=
3
-
4
LINE
40
22
4
-
-
-
-
-
1+0
-
-
-
-
-
-
8
-
23
First Total
293
113
14
-
1
8
6
4
1
6
7
8
9
-
-
-
-
2+3
Add to Reduce
2+9+3
1+1+3
1+4
-
-
-
-
-
-
-
-
-
-
-
-
8
-
5
Second Total
14
5
5
-
1
8
6
4
1
6
7
8
9
-
-
-
-
-
Reduce to Deduce
1+4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
5
Essence of Number
5
5
5
-
1
8
6
4
1
6
7
8
9

 

TWENTY ONE CENTIMETER LINE

 

T
=
2
-
9
TWENTYONE
141
42
6
-
1
2
3
4
5
6
7
8
9
C
=
3
-
10
CENTIMETER
112
49
4
-
-
-
-
-
-
-
-
-
-
L
=
3
-
4
LINE
40
22
4
-
-
-
-
-
-
-
-
-
-
-
-
-5
-
23
First Total
293
113
14
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
T
=
2
1
1
T
20
2
2
-
1
2
-
-
-
-
-
8
-
T
=
2
5
1
T
20
2
2
-
1
2
-
-
-
-
-
8
-
T
=
2
13
1
T
20
2
2
-
1
2
-
-
-
-
-
8
-
T
=
2
17
1
T
20
2
2
-
1
2
-
-
-
-
-
8
-
C
=
3
10
1
C
3
3
3
-
1
-
3
-
-
-
-
8
-
L
=
3
10
1
L
12
3
3
-
1
-
3
-
-
-
-
8
-
M
=
4
15
1
M
13
4
4
-
1
-
-
4
-
-
-
8
-
W
=
5
2
1
W
23
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
3
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
N
=
5
4
1
N
14
5
5
-
1
-
-
-
5
-
-
8
-
N
=
5
8
1
N
14
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
9
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
11
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
N
=
5
12
1
N
14
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
16
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
18
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
N
=
5
12
1
N
14
5
5
-
1
-
-
-
5
-
-
8
-
E
=
5
18
1
E
5
5
5
-
1
-
-
-
5
-
-
8
-
O
=
6
7
1
O
15
6
6
-
1
-
-
-
-
6
-
8
-
Y
=
7
6
1
Y
25
7
7
-
1
-
-
-
-
-
7
8
-
I
=
9
14
1
I
9
9
9
-
1
-
-
-
-
-
-
8
9
R
=
9
19
1
R
18
9
9
-
1
-
-
-
-
-
-
8
9
I
=
9
14
1
I
9
9
9
-
1
-
-
-
-
-
-
8
9
-
-
-
-
-
-
-
-
-
-
1
8
6
4
55
6
7
8
27
T
=
2
-
9
TWENTYONE
141
42
6
-
-
-
-
-
5+5
-
-
-
2+7
C
=
3
-
10
CENTIMETER
112
49
4
-
1
8
6
4
10
6
7
8
9
L
=
3
-
4
LINE
40
22
4
-
-
-
-
-
1+0
-
-
-
-
-
-
8
-
23
First Total
293
113
14
-
1
8
6
4
1
6
7
8
9
-
-
-
-
2+3
Add to Reduce
2+9+3
1+1+3
1+4
-
-
-
-
-
-
-
-
-
-
-
-
8
-
5
Second Total
14
5
5
-
1
8
6
4
1
6
7
8
9
-
-
-
-
-
Reduce to Deduce
1+4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
-
5
Essence of Number
5
5
5
-
1
8
6
4
1
6
7
8
9

 

TWENTY ONE CENTIMETER LINE

 

 

-
-
-
-
10
EXPERIENCE
149
77
5
-
1
2
3
4
5
6
7
8
9
E
=
5
1
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
X
=
6
2
1
X
24
6
6
-
-
-
-
-
-
6
-
-
-
P
=
7
3
1
P
16
7
7
-
-
-
-
-
-
-
7
-
-
E
=
5
4
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
E
=
5
7
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
8
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
C
=
3
9
1
C
3
3
3
-
-
-
3
-
-
-
-
-
-
E
=
5
10
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
59
-
10
EXPERIENCE
104
59
59
-
1
2
3
4
25
6
7
8
18
-
-
5+9
-
1+0
-
1+0+4
5+9
5+9
-
-
-
-
-
2+5
-
-
-
1+8
-
-
14
-
1
EXPERIENCE
5
14
14
-
1
2
3
4
7
6
7
8
9
-
-
1+4
-
-
1
-
1+4
1+4
-
-
-
-
-
-
-
-
-
-
-
-
5
-
1
EXPERIENCE
5
5
5
-
1
2
3
4
7
6
7
8
9

 

 

-
-
-
-
10
EXPERIENCE
149
77
5
-
1
2
3
4
5
6
7
8
9
E
=
5
1
1
E
5
5
5
-
1
2
-
4
5
-
-
8
-
X
=
6
2
1
X
24
6
6
-
1
2
-
4
-
6
-
8
-
P
=
7
3
1
P
16
7
7
-
1
2
-
4
-
-
7
8
-
E
=
5
4
1
E
5
5
5
-
1
2
-
4
5
-
-
8
-
R
=
9
5
1
R
18
9
9
-
1
2
-
4
-
-
-
8
9
I
=
9
6
1
I
9
9
9
-
1
2
-
4
-
-
-
8
9
E
=
5
7
1
E
5
5
5
-
1
2
-
4
5
-
-
8
-
N
=
5
8
1
N
14
5
5
-
1
2
-
4
5
-
-
8
-
C
=
3
9
1
C
3
3
3
-
1
2
3
4
-
-
-
8
-
E
=
5
10
1
E
5
5
5
-
1
2
-
4
5
-
-
8
-
-
-
59
-
10
EXPERIENCE
104
59
59
-
1
2
3
4
25
6
7
8
18
-
-
5+9
-
1+0
-
1+0+4
5+9
5+9
-
-
-
-
-
2+5
-
-
-
1+8
-
-
14
-
1
EXPERIENCE
5
14
14
-
1
2
3
4
7
6
7
8
9
-
-
1+4
-
-
1
-
1+4
1+4
-
-
-
-
-
-
-
-
-
-
-
-
5
-
1
EXPERIENCE
5
5
5
-
1
2
3
4
7
6
7
8
9

 

 

-
-
-
-
10
EXPERIENCE
149
77
5
-
1
2
3
4
5
6
7
8
9
C
=
3
9
1
C
3
3
3
-
1
2
3
4
-
-
-
8
-
E
=
5
1
1
E
5
5
5
-
1
2
-
4
5
-
-
8
-
E
=
5
4
1
E
5
5
5
-
1
2
-
4
5
-
-
8
-
E
=
5
7
1
E
5
5
5
-
1
2
-
4
5
-
-
8
-
N
=
5
8
1
N
14
5
5
-
1
2
-
4
5
-
-
8
-
E
=
5
10
1
E
5
5
5
-
1
2
-
4
5
-
-
8
-
X
=
6
2
1
X
24
6
6
-
1
2
-
4
-
6
-
8
-
P
=
7
3
1
P
16
7
7
-
1
2
-
4
-
-
7
8
-
R
=
9
5
1
R
18
9
9
-
1
2
-
4
-
-
-
8
9
I
=
9
6
1
I
9
9
9
-
1
2
-
4
-
-
-
8
9
-
-
59
-
10
EXPERIENCE
104
59
59
-
1
2
3
4
25
6
7
8
18
-
-
5+9
-
1+0
-
1+0+4
5+9
5+9
-
-
-
-
-
2+5
-
-
-
1+8
-
-
14
-
1
EXPERIENCE
5
14
14
-
1
2
3
4
7
6
7
8
9
-
-
1+4
-
-
1
-
1+4
1+4
-
-
-
-
-
-
-
-
-
-
-
-
5
-
1
EXPERIENCE
5
5
5
-
1
2
3
4
7
6
7
8
9

 

LETTERS RE-ARRANGED IN NUMERICAL ORDER

 

-
-
-
-
10
EXPERIENCE
149
77
5
-
3
5
6
7
9
C
=
3
9
1
C
3
3
3
-
3
-
-
-
-
E
=
5
1
1
E
5
5
5
-
-
5
-
-
-
E
=
5
4
1
E
5
5
5
-
-
5
-
-
-
E
=
5
7
1
E
5
5
5
-
-
5
-
-
-
N
=
5
8
1
N
14
5
5
-
-
5
-
-
-
E
=
5
10
1
E
5
5
5
-
-
5
-
-
-
X
=
6
2
1
X
24
6
6
-
-
-
6
-
-
P
=
7
3
1
P
16
7
7
-
-
-
-
7
-
R
=
9
5
1
R
18
9
9
-
-
-
-
-
9
I
=
9
6
1
I
9
9
9
-
-
-
-
-
9
-
-
59
-
10
EXPERIENCE
104
59
59
-
3
25
6
7
18
-
-
5+9
-
1+0
-
1+0+4
5+9
5+9
-
-
2+5
-
-
1+8
-
-
14
-
1
EXPERIENCE
5
14
14
-
3
7
6
7
9
-
-
1+4
-
-
1
-
1+4
1+4
-
-
-
-
-
-
-
-
5
-
1
EXPERIENCE
5
5
5
-
3
7
6
7
9

 

 

PROBLEMS - PROBLEMS

 
8
PROBLEMS
100
46
1
-
1
2
3
4
5
6
7
8
9
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
-
7
-
-
R
=
9
2
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
O
=
6
3
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
B
=
2
4
1
B
2
2
2
-
-
2
-
-
-
-
-
-
-
L
=
3
5
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
E
=
5
6
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
M
=
4
7
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
S
=
1
8
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
-
-
37
-
8
PROBLEMS
100
46
37
-
1
2
3
4
5
6
7
8
9
-
-
3+7
-
-
-
1+0+0
4+6
3+7
-
-
-
-
-
-
-
-
-
-
-
-
10
-
8
PROBLEMS
1
10
10
-
1
2
3
4
5
6
7
8
9
-
-
1+0
-
-
-
-
1+0
1+0
-
-
-
-
-
-
-
-
-
-
-
-
1
-
8
PROBLEMS
1
1
1
-
1
2
3
4
5
6
7
8
9

 

 
8
PROBLEMS
100
46
1
-
1
2
3
4
5
6
7
8
9
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
-
7
8
-
R
=
9
2
1
R
18
9
9
-
-
-
-
-
-
-
-
8
9
O
=
6
3
1
O
15
6
6
-
-
-
-
-
-
6
-
8
-
B
=
2
4
1
B
2
2
2
-
-
2
-
-
-
-
-
8
-
L
=
3
5
1
L
12
3
3
-
-
-
3
-
-
-
-
8
-
E
=
5
6
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
M
=
4
7
1
M
13
4
4
-
-
-
-
4
-
-
-
8
-
S
=
1
8
1
S
19
10
1
-
1
-
-
-
-
-
-
8
-
-
-
37
-
8
PROBLEMS
100
46
37
-
1
2
3
4
5
6
7
8
9
-
-
3+7
-
-
-
1+0+0
4+6
3+7
-
-
-
-
-
-
-
-
-
-
-
-
10
-
8
PROBLEMS
1
10
10
-
1
2
3
4
5
6
7
8
9
-
-
1+0
-
-
-
-
1+0
1+0
-
-
-
-
-
-
-
-
-
-
-
-
1
-
8
PROBLEMS
1
1
1
-
1
2
3
4
5
6
7
8
9

 

 
8
PROBLEMS
100
46
1
-
1
2
3
4
5
6
7
8
9
S
=
1
8
1
S
19
10
1
-
1
-
-
-
-
-
-
8
-
B
=
2
4
1
B
2
2
2
-
-
2
-
-
-
-
-
8
-
L
=
3
5
1
L
12
3
3
-
-
-
3
-
-
-
-
8
-
M
=
4
7
1
M
13
4
4
-
-
-
-
4
-
-
-
8
-
E
=
5
6
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
O
=
6
3
1
O
15
6
6
-
-
-
-
-
-
6
-
8
-
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
-
7
8
-
R
=
9
2
1
R
18
9
9
-
-
-
-
-
-
-
-
8
9
-
-
37
-
8
PROBLEMS
100
46
37
-
1
2
3
4
5
6
7
8
9
-
-
3+7
-
-
-
1+0+0
4+6
3+7
-
-
-
-
-
-
-
-
-
-
-
-
10
-
8
PROBLEMS
1
10
10
-
1
2
3
4
5
6
7
8
9
-
-
1+0
-
-
-
-
1+0
1+0
-
-
-
-
-
-
-
-
-
-
-
-
1
-
8
PROBLEMS
1
1
1
-
1
2
3
4
5
6
7
8
9

 

LETTERS RE-ARRANGED IN NUMERICAL ORDER

 
8
PROBLEMS
100
46
1
-
1
2
3
4
5
6
7
9
S
=
1
8
1
S
19
10
1
-
1
-
-
-
-
-
-
-
B
=
2
4
1
B
2
2
2
-
-
2
-
-
-
-
-
-
L
=
3
5
1
L
12
3
3
-
-
-
3
-
-
-
-
-
M
=
4
7
1
M
13
4
4
-
-
-
-
4
-
-
-
-
E
=
5
6
1
E
5
5
5
-
-
-
-
-
5
-
-
-
O
=
6
3
1
O
15
6
6
-
-
-
-
-
-
6
-
-
P
=
7
1
1
P
16
7
7
-
-
-
-
-
-
-
7
-
R
=
9
2
1
R
18
9
9
-
-
-
-
-
-
-
-
9
-
-
37
-
8
PROBLEMS
100
46
37
-
1
2
3
4
5
6
7
9
-
-
3+7
-
-
-
1+0+0
4+6
3+7
-
-
-
-
-
-
-
-
-
-
-
10
-
8
PROBLEMS
1
10
10
-
1
2
3
4
5
6
7
9
-
-
1+0
-
-
-
-
1+0
1+0
-
-
-
-
-
-
-
-
-
-
-
1
-
8
PROBLEMS
1
1
1
-
1
2
3
4
5
6
7
9

 

SOLVE PROBLEMS SOLVE

 
7
SOLVING
98
44
8
-
1
2
3
4
5
6
7
8
9
8
PROBLEMS
100
46
37
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
1
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
O
=
6
2
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
L
=
3
3
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
V
=
4
4
1
V
22
4
4
-
-
-
-
4
-
-
-
-
-
I
=
9
5
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
N
=
5
6
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
G
=
7
7
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
-
-
35
-
-
-
98
44
35
-
-
-
-
-
-
-
-
-
-
P
=
7
8
1
P
16
7
7
-
-
-
-
-
-
-
7
-
-
R
=
9
9
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
O
=
6
10
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
B
=
2
11
1
B
2
2
2
-
-
2
-
-
-
-
-
-
-
L
=
3
12
1
L
12
3
3
-
-
-
3
-
-
-
-
-
-
E
=
5
13
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
M
=
4
14
1
M
13
4
4
-
-
-
-
4
-
-
-
-
-
S
=
1
15
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
-
-
37
-
-
-
100
46
37
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SOLVING
98
44
35
-
2
2
6
8
10
12
14
8
18
-
-
-
-
8
PROBLEMS
100
46
37
-
-
-
-
-
1+0
1+2
1+4
-
1+8
-
-
72
-
15
First Total
198
90
72
-
2
2
6
8
1
3
5
8
9
-
-
7+2
-
1+5
Add to Reduce
1+9+8
9+0
7+2
-
-
-
-
-
-
-
-
-
-
-
-
9
-
6
Second Total
18
9
9
-
2
2
6
8
1
3
5
8
9
-
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
6
Essence of Number
9
9
9
-
2
2
6
8
1
3
5
8
9

 

 
7
SOLVING
98
44
8
-
1
2
3
4
5
6
7
8
9
8
PROBLEMS
100
46
37
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
1
1
S
19
10
1
-
1
-
-
-
-
-
-
8
-
O
=
6
2
1
O
15
6
6
-
-
-
-
-
-
6
-
8
-
L
=
3
3
1
L
12
3
3
-
-
-
3
-
-
-
-
8
-
V
=
4
4
1
V
22
4
4
-
-
-
-
4
-
-
-
8
-
I
=
9
5
1
I
9
9
9
-
-
-
-
-
-
-
-
8
9
N
=
5
6
1
N
14
5
5
-
-
-
-
-
5
-
-
8
-
G
=
7
7
1
G
7
7
7
-
-
-
-
-
-
-
7
8
-
P
=
7
8
1
P
16
7
7
-
-
-
-
-
-
-
7
8
-
R
=
9
9
1
R
18
9
9
-
-
-
-
-
-
-
-
8
9
O
=
6
10
1
O
15
6
6
-
-
-
-
-
-
6
-
8
-
B
=
2
11
1
B
2
2
2
-
-
2
-
-
-
-
-
8
-
L
=
3
12
1
L
12
3
3
-
-
-
3
-
-
-
-
8
-
E
=
5
13
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
M
=
4
14
1
M
13
4
4
-
-
-
-
4
-
-
-
8
-
S
=
1
15
1
S
19
10
1
-
1
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SOLVING
98
44
35
-
2
2
6
8
10
12
14
8
18
-
-
-
-
8
PROBLEMS
100
46
37
-
-
-
-
-
1+0
1+2
1+4
-
1+8
-
-
72
-
15
First Total
198
90
72
-
2
2
6
8
1
3
5
8
9
-
-
7+2
-
1+5
Add to Reduce
1+9+8
9+0
7+2
-
-
-
-
-
-
-
-
-
-
-
-
9
-
6
Second Total
18
9
9
-
2
2
6
8
1
3
5
8
9
-
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
6
Essence of Number
9
9
9
-
2
2
6
8
1
3
5
8
9

 

 
7
SOLVING
98
44
8
-
1
2
3
4
5
6
7
8
9
8
PROBLEMS
100
46
37
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
8
9
S
=
1
1
1
S
19
10
1
-
1
-
-
-
-
-
-
8
-
S
=
1
15
1
S
19
10
1
-
1
-
-
-
-
-
-
8
-
B
=
2
11
1
B
2
2
2
-
-
2
-
-
-
-
-
8
-
L
=
3
3
1
L
12
3
3
-
-
-
3
-
-
-
-
8
-
L
=
3
12
1
L
12
3
3
-
-
-
3
-
-
-
-
8
-
V
=
4
4
1
V
22
4
4
-
-
-
-
4
-
-
-
8
-
M
=
4
14
1
M
13
4
4
-
-
-
-
4
-
-
-
8
-
N
=
5
6
1
N
14
5
5
-
-
-
-
-
5
-
-
8
-
E
=
5
13
1
E
5
5
5
-
-
-
-
-
5
-
-
8
-
O
=
6
2
1
O
15
6
6
-
-
-
-
-
-
6
-
8
-
O
=
6
10
1
O
15
6
6
-
-
-
-
-
-
6
-
8
-
G
=
7
7
1
G
7
7
7
-
-
-
-
-
-
-
7
8
-
P
=
7
8
1
P
16
7
7
-
-
-
-
-
-
-
7
8
-
I
=
9
5
1
I
9
9
9
-
-
-
-
-
-
-
-
8
9
R
=
9
9
1
R
18
9
9
-
-
-
-
-
-
-
-
8
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SOLVING
98
44
35
-
2
2
6
8
10
12
14
8
18
-
-
-
-
8
PROBLEMS
100
46
37
-
-
-
-
-
1+0
1+2
1+4
-
1+8
-
-
72
-
15
First Total
198
90
72
-
2
2
6
8
1
3
5
8
9
-
-
7+2
-
1+5
Add to Reduce
1+9+8
9+0
7+2
-
-
-
-
-
-
-
-
-
-
-
-
9
-
6
Second Total
18
9
9
-
2
2
6
8
1
3
5
8
9
-
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
6
Essence of Number
9
9
9
-
2
2
6
8
1
3
5
8
9

 

LETTERS RE-ARRANGED IN NUMERICAL ORDER

 
7
SOLVING
98
44
8
-
1
2
3
4
5
6
7
9
8
PROBLEMS
100
46
37
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
2
3
4
5
6
7
9
S
=
1
1
1
S
19
10
1
-
1
-
-
-
-
-
-
-
S
=
1
15
1
S
19
10
1
-
1
-
-
-
-
-
-
-
B
=
2
11
1
B
2
2
2
-
-
2
-
-
-
-
-
-
L
=
3
3
1
L
12
3
3
-
-
-
3
-
-
-
-
-
L
=
3
12
1
L
12
3
3
-
-
-
3
-
-
-
-
-
V
=
4
4
1
V
22
4
4
-
-
-
-
4
-
-
-
-
M
=
4
14
1
M
13
4
4
-
-
-
-
4
-
-
-
-
N
=
5
6
1
N
14
5
5
-
-
-
-
-
5
-
-
-
E
=
5
13
1
E
5
5
5
-
-
-
-
-
5
-
-
-
O
=
6
2
1
O
15
6
6
-
-
-
-
-
-
6
-
-
O
=
6
10
1
O
15
6
6
-
-
-
-
-
-
6
-
-
G
=
7
7
1
G
7
7
7
-
-
-
-
-
-
-
7
-
P
=
7
8
1
P
16
7
7
-
-
-
-
-
-
-
7
-
I
=
9
5
1
I
9
9
9
-
-
-
-
-
-
-
-
9
R
=
9
9
1
R
18
9
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SOLVING
98
44
35
-
2
2
6
8
10
12
14
18
-
-
-
-
8
PROBLEMS
100
46
37
-
-
-
-
-
1+0
1+2
1+4
1+8
-
-
72
-
15
First Total
198
90
72
-
2
2
6
8
1
3
5
9
-
-
7+2
-
1+5
Add to Reduce
1+9+8
9+0
7+2
-
-
-
-
-
-
-
-
-
-
-
9
-
6
Second Total
18
9
9
-
2
2
6
8
1
3
5
9
-
-
-
-
-
Reduce to Deduce
1+8
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
6
Essence of Number
9
9
9
-
2
2
6
8
1
3
5
9

 

 

-
-
-
-
-
THE CODING
-
-
-
T
=
2
-
3
THE
33
15
6
C
=
3
-
6
CODING
52
34
7
-
-
5
-
9
THE CODING
85
49
13
-
-
-
-
-
-
8+5
4+9
1+3
-
-
5
-
9
THE CODING
13
13
4
-
-
-
-
-
-
1+3
1+3
-
-
-
5
-
9
THE CODING
4
4
4

 

 

-
-
-
-
-
THE CODING
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
-
-
-
-
C
=
3
-
6
CODING
52
34
7
-
-
-
-
-
-
-
-
-
-
-
-
5
-
9
THE CODING
85
49
13
-
1
2
3
4
5
6
7
8
9
T
=
2
1
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
H
=
8
2
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
E
=
5
3
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
15
-
-
THE CODING
33
15
15
-
-
-
-
-
-
-
-
-
-
C
=
3
4
1
C
3
3
3
-
-
-
3
-
-
-
-
-
-
O
=
6
5
1
O
15
6
6
-
-
-
-
-
-
6
-
-
-
D
=
4
6
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
I
=
9
7
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
N
=
5
8
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
G
=
7
9
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
-
-
34
-
-
THE CODING
52
34
34
-
-
-
-
-
-
-
-
-
-
T
=
2
-
3
THE
33
15
6
-
1
2
3
4
10
6
7
8
9
C
=
3
-
6
CODING
52
34
7
-
-
-
-
-
1+0
-
-
-
-
-
-
5
-
9
THE CODING
85
49
13
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
8+5
4+9
1+3
-
-
-
-
-
-
-
-
-
-
-
-
5
-
9
THE CODING
13
13
4
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
9
THE CODING
4
4
4
-
1
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
THE CODING
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
-
-
-
-
C
=
3
-
6
CODING
52
34
7
-
-
-
-
-
-
-
-
-
-
-
-
5
-
9
THE CODING
85
49
13
-
1
2
3
4
5
6
7
8
9
T
=
2
1
1
T
20
2
2
-
1
2
-
-
-
-
-
-
-
H
=
8
2
1
H
8
8
8
-
1
-
-
-
-
-
-
8
-
E
=
5
3
1
E
5
5
5
-
1
-
-
-
5
-
-
-
-
C
=
3
4
1
C
3
3
3
-
1
-
3
-
-
-
-
-
-
O
=
6
5
1
O
15
6
6
-
1
-
-
-
-
6
-
-
-
D
=
4
6
1
D
4
4
4
-
1
-
-
4
-
-
-
-
-
I
=
9
7
1
I
9
9
9
-
1
-
-
-
-
-
-
-
9
N
=
5
8
1
N
14
5
5
-
1
-
-
-
5
-
-
-
-
G
=
7
9
1
G
7
7
7
-
1
-
-
-
-
-
7
-
-
T
=
2
-
3
THE
33
15
6
-
1
2
3
4
10
6
7
8
9
C
=
3
-
6
CODING
52
34
7
-
-
-
-
-
1+0
-
-
-
-
-
-
5
-
9
THE CODING
85
49
13
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
8+5
4+9
1+3
-
-
-
-
-
-
-
-
-
-
-
-
5
-
9
THE CODING
13
13
4
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
9
THE CODING
4
4
4
-
1
2
3
4
5
6
7
8
9

 

TRANSPOSED LETTERS REARRANGED IN NUMERICAL ORDER

 

-
-
-
-
-
THE CODING
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
-
-
-
-
C
=
3
-
6
CODING
52
34
7
-
-
-
-
-
-
-
-
-
-
-
-
5
-
9
THE CODING
85
49
13
-
1
2
3
4
5
6
7
8
9
T
=
2
1
1
T
20
2
2
-
1
2
-
-
-
-
-
-
-
C
=
3
4
1
C
3
3
3
-
1
-
3
-
-
-
-
-
-
D
=
4
6
1
D
4
4
4
-
1
-
-
4
-
-
-
-
-
E
=
5
3
1
E
5
5
5
-
1
-
-
-
5
-
-
-
-
N
=
5
8
1
N
14
5
5
-
1
-
-
-
5
-
-
-
-
O
=
6
5
1
O
15
6
6
-
1
-
-
-
-
6
-
-
-
G
=
7
9
1
G
7
7
7
-
1
-
-
-
-
-
7
-
-
H
=
8
2
1
H
8
8
8
-
1
-
-
-
-
-
-
8
-
I
=
9
7
1
I
9
9
9
-
1
-
-
-
-
-
-
-
9
T
=
2
-
3
THE
33
15
6
-
1
2
3
4
10
6
7
8
9
C
=
3
-
6
CODING
52
34
7
-
-
-
-
-
1+0
-
-
-
-
-
-
5
-
9
THE CODING
85
49
13
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
8+5
4+9
1+3
-
-
-
-
-
-
-
-
-
-
-
-
5
-
9
THE CODING
13
13
4
-
1
2
3
4
5
6
7
8
9
-
-
-
-
-
-
1+3
1+3
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
9
THE CODING
4
4
4
-
1
2
3
4
5
6
7
8
9

 

 

-
-
-
-
-
THE CODING
-
-
-
T
=
2
-
3
THE
33
15
6
C
=
3
-
6
CODING
52
34
7
-
-
5
-
9
THE CODING
85
49
13
-
-
-
-
-
-
8+5
4+9
1+3
-
-
5
-
9
THE CODING
13
13
4
-
-
-
-
-
-
1+3
1+3
-
-
-
5
-
9
THE CODING
4
4
4

 


The Discovery of the Expanding Universe - SDSS SkyServer

https://skyserver.sdss.org › astro › universe › universe

The universe is in a constant state of change. The expanding universe, a new idea based on modern physics, laid to rest the paradoxes that troubled astronomers ...

The Expanding Universe

For thousands of years, astronomers wrestled with basic questions about the size and age of the universe. Does the universe go on forever, or does it have an edge somewhere? Has it always existed, or did it come to being some time in the past? In 1929, Edwin Hubble, an astronomer at Caltech, made a critical discovery that soon led to scientific answers for these questions: he discovered that the universe is expanding.

The ancient Greeks recognized that it was difficult to imagine what an infinite universe might look like. But they also wondered that if the universe were finite, and you stuck out your hand at the edge, where would your hand go? The Greeks' two problems with the universe represented a paradox - the universe had to be either finite or infinite, and both alternatives presented problems.

After the rise of modern astronomy, another paradox began to puzzle astronomers. In the early 1800s, German astronomer Heinrich Olbers argued that the universe must be finite. If the Universe were infinite and contained stars throughout, Olbers said, then if you looked in any particular direction, your line-of-sight would eventually fall on the surface of a star. Although the apparent size of a star in the sky becomes smaller as the distance to the star increases, the brightness of this smaller surface remains a constant. Therefore, if the Universe were infinite, the whole surface of the night sky should be as bright as a star. Obviously, there are dark areas in the sky, so the universe must be finite.

But, when Isaac Newton discovered the law of gravity, he realized that gravity is always attractive. Every object in the universe attracts every other object. If the universe truly were finite, the attractive forces of all the objects in the universe should have caused the entire universe to collapse on itself. This clearly had not happened, and so astronomers were presented with a paradox.

When Einstein developed his theory of gravity in the General Theory of Relativity, he thought he ran into the same problem that Newton did: his equations said that the universe should be either expanding or collapsing, yet he assumed that the universe was static. His original solution contained a constant term, called the cosmological constant, which cancelled the effects of gravity on very large scales, and led to a static universe. After Hubble discovered that the universe was expanding, Einstein called the cosmological constant his "greatest blunder."

At around the same time, larger telescopes were being built that were able to accurately measure the spectra, or the intensity of light as a function of wavelength, of faint objects. Using these new data, astronomers tried to understand the plethora of faint, nebulous objects they were observing. Between 1912 and 1922, astronomer Vesto Slipher at the Lowell Observatory in Arizona discovered that the spectra of light from many of these objects was systematically shifted to longer wavelengths, or redshifted. A short time later, other astronomers showed that these nebulous objects were distant galaxies.

The Discovery of the Expanding Universe

Meanwhile, other physicists and mathematicians working on Einstein's theory of gravity discovered the equations had some solutions that described an expanding universe. In these solutions, the light coming from distant objects would be redshifted as it traveled through the expanding universe. The redshift would increase with increasing distance to the object.

Edwin Hubble

In 1929 Edwin Hubble, working at the Carnegie Observatories in Pasadena, California, measured the redshifts of a number of distant galaxies. He also measured their relative distances by measuring the apparent brightness of a class of variable stars called Cepheids in each galaxy. When he plotted redshift against relative distance, he found that the redshift of distant galaxies increased as a linear function of their distance. The only explanation for this observation is that the universe was expanding.

Once scientists understood that the universe was expanding, they immediately realized that it would have been smaller in the past. At some point in the past, the entire universe would have been a single point. This point, later called the big bang, was the beginning of the universe as we understand it today.

The expanding universe is finite in both time and space. The reason that the universe did not collapse, as Newton's and Einstein's equations said it might, is that it had been expanding from the moment of its creation. The universe is in a constant state of change. The expanding universe, a new idea based on modern physics, laid to rest the paradoxes that troubled astronomers from ancient times until the early 20th Century.

The Heavy Elements

Astronomers are not only interested in the fate of the universe; they are also interested in understanding its present physical state. One question they try to answer is why the universe is primarily composed of hydrogen and helium, and what is responsible for the relatively small concentration of the heavier elements.

With the rise of nuclear physics in the 1930s and 40s, scientists started to try to explain the abundances of heavier elements by assuming they were synthesized out of primordial hydrogen in the early universe. In the late 1940s, American physicists George Gamow, Robert Herman, and Ralph Alpher realized that long ago, the universe was much hotter and denser. They made calculations to show whether nuclear reactions that took place at those higher temperatures could have created the heavy elements.

Unfortunately, with the exception of helium, they found that it was impossible to form heavier elements in any appreciable quantity. Today, we understand that heavy elements were synthesized either in the cores of stars or during supernovae, when a large dying star implodes.

Gamow, Herman, and Alpher did realize, though, that if the universe were hotter and denser in the past, radiation should still be left over from the early universe. This radiation would have a well-defined spectrum (called a blackbody spectrum) that depends on its temperature. As the universe expanded, the spectrum of this light would have been redshifted to longer wavelengths, and the temperature associated with the spectrum would have decreased by a factor of over one thousand as the universe cooled.

 

-
-
-
-
20
THE EXPANDING UNIVERSE
-
-
-
T
=
2
-
3
THE
33
15
6
E
=
5
-
9
EXPANDING
94
49
4
U
=
3
-
8
UNIVERSE
113
41
5
-
-
10
-
20
THE EXPANDING UNIVERSE
240
105
15
-
-
1+0
-
2+0
-
2+4+0
1+0+5
1+5
-
-
1
-
2
THE EXPANDING UNIVERSE
6
6
6

 

 

-
-
-
-
20
THE EXPANDING UNIVERSE
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
-
-
-
-
E
=
5
-
9
EXPANDING
94
49
4
-
-
-
-
-
-
-
-
-
-
U
=
3
-
8
UNIVERSE
113
41
5
-
-
-
-
-
-
-
-
-
-
-
-
10
-
20
THE EXPANDING UNIVERSE
240
105
15
-
1
2
3
4
5
6
7
8
9
T
=
2
1
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
H
=
8
2
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
E
=
5
3
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
15
-
-
THE EXPANDING UNIVERSE
33
15
15
-
-
-
-
-
-
-
-
-
-
E
=
5
4
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
X
=
6
5
1
X
24
6
6
-
-
-
-
-
-
6
-
-
-
P
=
7
6
1
P
16
7
7
-
-
-
-
-
-
-
7
-
-
A
=
1
7
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
N
=
5
8
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
D
=
4
9
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
I
=
9
10
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
N
=
5
11
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
G
=
7
12
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
-
-
49
4
-
THE EXPANDING UNIVERSE
94
49
49
-
-
-
-
-
-
-
-
-
-
U
=
3
13
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
N
=
5
14
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
I
=
9
15
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
V
=
4
16
1
V
22
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
17
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
R
=
9
18
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
S
=
1
19
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
E
=
5
20
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
15
-
20
THE EXPANDING UNIVERSE
113
15
15
-
2
2
3
8
35
6
14
8
27
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
3+5
-
1+4
-
2+7
E
=
5
-
9
EXPANDING
94
34
7
-
2
2
3
8
8
6
5
8
9
U
=
3
-
8
UNIVERSE
113
15
6
-
-
-
-
-
-
-
-
-
-
-
-
10
-
20
THE EXPANDING UNIVERSE
240
49
13
-
2
2
3
8
8
6
5
8
9
-
-
1+0
-
2+0
-
2+4+0
4+9
1+3
-
-
-
-
-
-
-
-
-
-
-
-
1
-
2
THE EXPANDING UNIVERSE
6
13
4
-
2
2
3
8
8
6
5
8
9

 

 

-
-
-
-
20
THE EXPANDING UNIVERSE
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
-
-
-
-
E
=
5
-
9
EXPANDING
94
49
4
-
-
-
-
-
-
-
-
-
-
U
=
3
-
8
UNIVERSE
113
41
5
-
-
-
-
-
-
-
-
-
-
-
-
10
-
20
THE EXPANDING UNIVERSE
240
105
15
-
1
2
3
4
5
6
7
8
9
T
=
2
1
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
H
=
8
2
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
E
=
5
3
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
4
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
X
=
6
5
1
X
24
6
6
-
-
-
-
-
-
6
-
-
-
P
=
7
6
1
P
16
7
7
-
-
-
-
-
-
-
7
-
-
A
=
1
7
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
N
=
5
8
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
D
=
4
9
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
I
=
9
10
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
N
=
5
11
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
G
=
7
12
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
U
=
3
13
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
N
=
5
14
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
I
=
9
15
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
V
=
4
16
1
V
22
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
17
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
R
=
9
18
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
S
=
1
19
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
E
=
5
20
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
THE EXPANDING UNIVERSE
-
-
-
-
2
2
3
8
35
6
14
8
27
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
3+5
-
1+4
-
2+7
E
=
5
-
9
EXPANDING
94
34
7
-
2
2
3
8
8
6
5
8
9
U
=
3
-
8
UNIVERSE
113
15
6
-
-
-
-
-
-
-
-
-
-
-
-
10
-
20
THE EXPANDING UNIVERSE
240
49
13
-
2
2
3
8
8
6
5
8
9
-
-
1+0
-
2+0
-
2+4+0
4+9
1+3
-
-
-
-
-
-
-
-
-
-
-
-
1
-
2
THE EXPANDING UNIVERSE
6
13
4
-
2
2
3
8
8
6
5
8
9

 

TRANSPOSED LETTERS REARRANGED IN NUMERICAL ORDER

 

-
-
-
-
20
THE EXPANDING UNIVERSE
-
-
-
-
1
2
3
4
5
6
7
8
9
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
-
-
-
-
-
E
=
5
-
9
EXPANDING
94
49
4
-
-
-
-
-
-
-
-
-
-
U
=
3
-
8
UNIVERSE
113
41
5
-
-
-
-
-
-
-
-
-
-
-
-
10
-
20
THE EXPANDING UNIVERSE
240
105
15
-
1
2
3
4
5
6
7
8
9
A
=
1
7
1
A
1
1
1
-
1
-
-
-
-
-
-
-
-
S
=
1
19
1
S
19
10
1
-
1
-
-
-
-
-
-
-
-
T
=
2
1
1
T
20
2
2
-
-
2
-
-
-
-
-
-
-
U
=
3
13
1
U
21
3
3
-
-
-
3
-
-
-
-
-
-
D
=
4
9
1
D
4
4
4
-
-
-
-
4
-
-
-
-
-
V
=
4
16
1
V
22
4
4
-
-
-
-
4
-
-
-
-
-
E
=
5
3
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
4
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
8
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
11
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
N
=
5
14
1
N
14
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
17
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
E
=
5
20
1
E
5
5
5
-
-
-
-
-
5
-
-
-
-
X
=
6
5
1
X
24
6
6
-
-
-
-
-
-
6
-
-
-
P
=
7
6
1
P
16
7
7
-
-
-
-
-
-
-
7
-
-
G
=
7
12
1
G
7
7
7
-
-
-
-
-
-
-
7
-
-
H
=
8
2
1
H
8
8
8
-
-
-
-
-
-
-
-
8
-
I
=
9
10
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
I
=
9
15
1
I
9
9
9
-
-
-
-
-
-
-
-
-
9
R
=
9
18
1
R
18
9
9
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
THE EXPANDING UNIVERSE
-
-
-
-
2
2
3
8
35
6
14
8
27
T
=
2
-
3
THE
33
15
6
-
-
-
-
-
3+5
-
1+4
-
2+7
E
=
5
-
9
EXPANDING
94
34
7
-
2
2
3
8
8
6
5
8
9
U
=
3
-
8
UNIVERSE
113
15
6
-
-
-
-
-
-
-
-
-
-
-
-
10
-
20
THE EXPANDING UNIVERSE
240
49
13
-
2
2
3
8
8
6
5
8
9
-
-
1+0
-
2+0
-
2+4+0
4+9
1+3
-
-
-
-
-
-
-
-
-
-
-
-
1
-
2
THE EXPANDING UNIVERSE
6
13
4
-
2
2
3
8
8
6
5
8
9

 

 

'Our revels now are ended'

Our revels now are ended. These our actors,
As I foretold you, were all spirits and
Are melted into air, into thin air:
And, like the baseless fabric of this vision,
The cloud-capp'd towers, the gorgeous palaces,
The solemn temples, the great globe itself,
Yea, all which it inherit, shall dissolve
And, like this insubstantial pageant faded,
Leave not a rack behind. We are such stuff
As dreams are made on, and our little life
Is rounded with a sleep.

William Shakespeare
From The Tempest, Act 4 Scene 1

William Shakespeare ( 26 April 1564 (baptised) – 23 April 1616)
was an English poet and playwright, widely regarded as the greatest writer in the English ...

 

 

4
MASS
52
7
7
6
ENERGY
74
38
2
10
-
126
45
9
1+0
-
1+2+6
4+5
-
1
-
9
9
9

 

 

8
ELECTRIC
75
39
3
9
PARTICLES
103
40
4
17
First Total
178
79
7
1+7
Add to Reduce
1+7+8
7+9
-
8
Second Total
16
16
7
-
Reduce to Deduce
1+6
1+6
-
8
Essence of Number
7
7
7

 

 

5
GLUON
69
24
6
6
GLUONS
24
15
7

 

 

-
7
P
R
O
T
O
N
S
-
-
-
-
-
--
-
-
-
-
-
-
-
-
6
-
6
5
1
+
=
18
1+8
=
9
-
9
-
9
-
-
-
-
15
-
15
14
19
+
=
63
6+3
=
9
-
9
-
9
-
7
P
R
O
T
O
N
S
-
-
-
-
-
--
-
-
-
-
-
-
7
9
-
2
-
-
-
+
=
18
1+8
=
9
-
9
-
9
-
-
16
18
-
20
-
-
-
+
=
54
5+4
=
9
-
9
-
9
-
7
P
R
O
T
O
N
S
-
-
-
-
-
--
-
-
-
-
-
-
16
18
15
20
15
14
19
+
=
117
1+1+7
=
9
-
9
-
9
-
-
7
9
6
2
6
5
1
+
=
36
3+6
=
9
-
9
-
9
-
7
P
R
O
T
O
N
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
1
=
1
-
1
-
-
-
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
-
2
3
-
-
--
--
--
--
--
--
-
-
3
-
-
-
-
-
-
-
4
-
-
--
--
--
--
--
--
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
5
occurs
x
1
=
5
-
5
-
-
-
-
6
-
6
-
-
-
-
6
occurs
x
2
=
12
1+2
3
-
-
7
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
-
7
8
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
9
occurs
x
1
=
9
-
9
15
7
P
R
O
T
O
N
S
-
-
30
-
-
7
-
36
-
27
1+5
-
-
9
-
-
-
-
-
-
-
3+0
-
-
-
-
3+6
-
2+7
6
7
P
R
O
T
O
N
S
-
-
3
-
-
7
-
9
-
9
-
-
7
9
6
2
6
5
9
-
-
-
-
-
-
-
-
-
-
6
7
P
R
O
T
O
N
S
-
-
3
-
-
7
-
9
-
9

 

 

7
P
R
O
T
O
N
S
-
-
-
-
-
--
-
-
-
-
-
-
-
6
-
6
5
1
+
=
18
1+8
=
9
-
9
-
9
-
-
-
15
-
15
14
19
+
=
63
6+3
=
9
-
9
-
9
7
P
R
O
T
O
N
S
-
-
-
-
-
--
-
-
-
-
-
7
9
-
2
-
-
-
+
=
18
1+8
=
9
-
9
-
9
-
16
18
-
20
-
-
-
+
=
54
5+4
=
9
-
9
-
9
7
P
R
O
T
O
N
S
-
-
-
-
-
--
-
-
-
-
-
16
18
15
20
15
14
19
+
=
117
1+1+7
=
9
-
9
-
9
-
7
9
6
2
6
5
1
+
=
36
3+6
=
9
-
9
-
9
7
P
R
O
T
O
N
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
1
=
1
-
1
-
-
-
-
2
-
-
-
-
-
2
occurs
x
1
=
2
-
2
-
-
-
-
-
-
5
-
-
-
5
occurs
x
1
=
5
-
5
-
-
-
6
-
6
-
-
-
-
6
occurs
x
2
=
12
1+2
3
-
7
-
-
-
-
-
-
-
-
7
occurs
x
1
=
7
-
7
-
-
9
-
-
-
-
-
-
-
9
occurs
x
1
=
9
-
9
7
P
R
O
T
O
N
S
-
-
30
-
-
7
-
36
-
27
-
-
9
-
-
-
-
-
-
-
3+0
-
-
-
-
3+6
-
2+7
7
P
R
O
T
O
N
S
-
-
3
-
-
7
-
9
-
9
-
7
9
6
2
6
5
9
-
-
-
-
-
-
-
-
-
-
7
P
R
O
T
O
N
S
-
-
3
-
-
7
-
9
-
9

 

 

-
8
N
E
U
T
R
O
N
S
-
-
-
-
-
--
-
-
-
-
-
-
5
-
-
-
-
6
5
1
+
=
17
1+7
=
8
-
8
-
8
-
-``
14
-
-
-
-
15
14
19
+
=
62
6+2
=
8
-
8
-
8
-
8
N
E
U
T
R
O
N
S
-
-
-
-
-
-
-
-
-
-
-
-
-
5
3
2
9
-
-
-
+
=
19
1+9
=
10
1+0
1
-
1
-
-
-
5
3
20
18
-
-
-
+
=
64
6+4
=
10
1+0
1
-
1
-
8
N
E
U
T
R
O
N
S
-
-
-
-
-
--
-
-
-
-
-
-
14
5
21
20
18
15
14
19
+
=
126
1+2+6
=
9
-
9
-
9
-
-
5
5
3
2
9
6
5
1
+
=
36
3+6
=
9
-
9
-
9
-
8
N
E
U
T
R
O
N
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
1
=
1
-
1
-
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
1
=
2
-
2
-
-
-
-
3
-
-
-
-
-
-
-
3
occurs
x
1
=
3
-
3
4
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
5
5
-
-
-
-
5
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
-
6
-
-
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
=-
-
-
9
-
-
-
-
-
9
occurs
x
1
=
9
-
9
19
8
N
E
U
T
R
O
N
S
-
-
26
-
-
8
-
36
-
27
1+9
-
--
-
-
-
9
-
-
-
-
-
2+6
-
-
-
-
3+6
-
2+7
10
8
N
E
U
T
R
O
N
S
-
-
8
-
-
8
-
9
-
9
1+0
-
5
5
3
2
9
6
5
1
-
-
-
-
-
-
-
-
-
-
1
8
N
E
U
T
R
O
N
S
-
-
8
-
-
8
-
9
-
9

 

 

8
N
E
U
T
R
O
N
S
-
-
-
-
-
--
-
-
-
-
-
5
-
-
-
-
6
5
1
+
=
17
1+7
=
8
-
8
-
8
-``
14
-
-
-
-
15
14
19
+
=
62
6+2
=
8
-
8
-
8
8
N
E
U
T
R
O
N
S
-
-
-
-
-
-
-
-
-
-
-
-
5
3
2
9
-
-
-
+
=
19
1+9
=
10
1+0
1
-
1
-
-
5
3
20
18
-
-
-
+
=
64
6+4
=
10
1+0
1
-
1
8
N
E
U
T
R
O
N
S
-
-
-
-
-
--
-
-
-
-
-
14
5
21
20
18
15
14
19
+
=
126
1+2+6
=
9
-
9
-
9
-
5
5
3
2
9
6
5
1
+
=
36
3+6
=
9
-
9
-
9
8
N
E
U
T
R
O
N
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
1
=
1
-
1
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
1
=
2
-
2
-
-
-
3
-
-
-
-
-
-
-
3
occurs
x
1
=
3
-
3
-
5
5
-
-
-
-
5
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
6
-
-
-
-
6
occurs
x
1
=
6
=
6
-
-
=-
-
-
9
-
-
-
-
-
9
occurs
x
1
=
9
-
9
8
N
E
U
T
R
O
N
S
-
-
26
-
-
8
-
36
-
27
-
--
-
-
-
9
-
-
-
-
-
2+6
-
-
-
-
3+6
-
2+7
8
N
E
U
T
R
O
N
S
-
-
8
-
-
8
-
9
-
9
-
5
5
3
2
9
6
5
1
-
-
-
-
-
-
-
-
-
-
8
N
E
U
T
R
O
N
S
-
-
8
-
-
8
-
9
-
9

 

 

-
9
E
L
E
C
T
R
O
N
S
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
6
5
1
+
=
12
1+2
=
3
-
3
-
3
-
-``
-
-
-
-
-
-
15
14
19
+
=
48
4+8
=
12
1+2
3
-
3
-
9
E
L
E
C
T
R
O
N
S
-
-
-
-
-
-
-
-
-
-
-
-
5
3
5
3
2
9
-
-
-
+
=
27
2+7
=
9
-
9
-
9
-
-
5
3
5
3
20
18
-
-
-
+
=
63
6+3
=
9
-
9
-
9
-
9
E
L
E
C
T
R
O
N
S
-
-
-
-
-
--
-
-
-
-
-
-
5
12
5
3
20
18
15
14
19
+
=
111
1+1+1
=
3
-
3
-
3
-
-
5
3
5
3
2
9
6
5
1
+
=
39
3+9
=
12
1+2
3
-
3
-
9
E
L
E
C
T
R
O
N
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
1
=
1
-
1
-
-
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
1
=
2
-
2
-
-
-
3
-
3
-
-
-
-
-
-
-
3
occurs
x
2
=
6
-
6
4
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
5
-
5
-
-
-
-
5
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
-
-
6
-
-
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
=-
-
=-
-
-
9
-
-
-
-
-
9
occurs
x
1
=
9
-
9
19
9
E
L
E
C
T
R
O
N
S
-
-
26
-
-
9
-
39
-
30
1+9
-
-
-
-
-
-
9
-
-
-
-
-
2+6
-
-
-
-
3+9
-
3+0
10
9
E
L
E
C
T
R
O
N
S
-
-
8
-
-
9
-
12
-
3
1+0
-
5
3
5
3
2
9
6
5
1
-
-
-
-
-
-
-
1+2
-
-
1
8
E
L
E
C
T
R
O
N
S
-
-
8
-
-
8
-
9
-
9

 

 

9
E
L
E
C
T
R
O
N
S
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
6
5
1
+
=
12
1+2
=
3
-
3
-
3
-``
-
-
-
-
-
-
15
14
19
+
=
48
4+8
=
12
1+2
3
-
3
9
E
L
E
C
T
R
O
N
S
-
-
-
-
-
-
-
-
-
-
-
5
3
5
3
2
9
-
-
-
+
=
27
2+7
=
9
-
9
-
9
-
5
3
5
3
20
18
-
-
-
+
=
63
6+3
=
9
-
9
-
9
9
E
L
E
C
T
R
O
N
S
-
-
-
-
-
--
-
-
-
-
-
5
12
5
3
20
18
15
14
19
+
=
111
1+1+1
=
3
-
3
-
3
-
5
3
5
3
2
9
6
5
1
+
=
39
3+9
=
12
1+2
3
-
3
9
E
L
E
C
T
R
O
N
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
1
=
1
-
1
-
-
-
-
-
2
-
-
-
-
-
-
2
occurs
x
1
=
2
-
2
-
-
3
-
3
-
-
-
-
-
-
-
3
occurs
x
2
=
6
-
6
-
5
-
5
-
-
-
-
5
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
-
6
-
-
-
-
6
occurs
x
1
=
6
=
6
-
=-
-
=-
-
-
9
-
-
-
-
-
9
occurs
x
1
=
9
-
9
9
E
L
E
C
T
R
O
N
S
-
-
26
-
-
9
-
39
-
30
-
-
-
-
-
-
9
-
-
-
-
-
2+6
-
-
-
-
3+9
-
3+0
9
E
L
E
C
T
R
O
N
S
-
-
8
-
-
9
-
12
-
3
-
5
3
5
3
2
9
6
5
1
-
-
-
-
-
-
-
1+2
-
-
8
E
L
E
C
T
R
O
N
S
-
-
8
-
-
8
-
9
-
9

 

 

G
=
7
5
GRAND
44
26
8
U
=
3
7
UNIFIED
68
41
5
T
=
2
6
THEORY
91
37
1

 

 

3
THE
33
15
6
8
PARTICLE
84
39
3

 

 

11
FUNDAMENTAL
111
39
3
5
FORCE
47
29
2
6
FORCES
66
30
3

 

 

7
ELECTRO
78
33
6
9
MAGNETISM
101
38
2

 

 

7
ELECTRO
78
33
6
8
MAGNETIC
72
36
9

 

 

6
STRONG
93
30
3
7
NUCLEAR
74
29
2
5
FORCE
47
29
2
18
First Total
214
88
7
1+8
Add to Reduce
2+1+4
8+8
-
9
Second Total
7
16
7
-
Reduce to Deduce
-
1+6
-
9
Essence of Number
7
7
7

 

 

4
WEAK
40
13
4
7
NUCLEAR
74
29
2
5
FORCE
47
29
2
16
Add to Reduce
161
71
8
1+6
Reduce to Deduce
1+6+1
7+1
-
7
Essence of Number
8
8
8

 

 

7
GRAVITY
102
39
3

 

 

7
UNIFIED
68
41
5

 

 

3
THE
33
15
6
5
GRAND
44
26
8
6
THEORY
91
37
1
2
OF
21
12
3
10
EVERYTHING
133
61
7
26
Add to Reduce
322
151
25
2+6
Reduce to Deduce
3+2+2
1+5+1
2+5
8
Essence of Number
7
7
7

 

 

3
THE
33
15
6
6
THEORY
91
37
1
2
OF
21
12
3
10
EVERYTHING
133
61
7
21
First Total
278
125
17
2+1
Add to Reduce
2+7+8
1+2+5
1+7
3
Second Total
17
8
8
-
Reduce to Deduce
1+7
-
-
3
Essence of Number
8
8
8

 

 

4
NANAO
44
17
8
9
NANO METER
105
42
6
5
METER
61
25
7

 

 

8
TRILLION
109
46
1
8
ELECTRON
92
38
2
5
VOLTS
88
16
7

 

 

F
=
6
4
FIRE
38
29
2
L
=
3
5
LIGHT
56
29
2

 

 

7
ELEMENT
74
29
2

 

 

10
ELEMENTARY
118
46
1
9
PARTICLES
103
40
4

 

 

6
ATOMIC
61
25
7
6
ENERGY
74
38
2
12
Add to Reduce
135
63
9
1+2
Reduce to Deduce
1+3+5
6+3
-
3
Essence of Number
9
9
9

 

 

9
TERASCALE
84
30
3

 

 

4
LIFE
32
23
5
4
FILE
32
23
5

 

 

E
=
5
-
2
EX
11
2
2
U
=
3
-
6
UMBRIS
82
28
1
E
=
5
-
2
ET
25
7
7
I
=
9
-
10
IMAGINIBUS
104
50
5
I
=
9
-
2
IN
23
14
5
V
=
4
-
9
VERITATEM
113
41
5
-
-
35
-
31
First Total
358
142
25
-
-
3+5
-
3+1
Add to Reduce
3+5+8
1+4+2
2+5
-
-
8
-
4
Second Total
16
7
7
-
-
-
-
-
Reduce to Deduce
1+6
-
-
-
-
8
-
4
Essence of Number
7
7
7

 

 

O
=
6
-
3
OUT
56
11
2
O
=
6
-
2
OF
21
12
3
S
=
1
-
7
SHADOWS
89
26
8
A
=
1
-
3
AND
82
28
1
P
=
7
-
9
PHANTASMS
111
30
3
I
=
9
-
4
INTO
58
22
4
T
=
2
-
5
TRUTH
87
24
6
-
-
32
-
33
Add to Reduce
441
135
27
-
-
3+2
-
3+3
Reduce to Deduce
4+4+1
1+3+5
2+7
-
-
5
-
6
Essence of Number
9
9
9

 

 

3
THE
33
15
6
7
THUNDER
90
36
9
4
BOLT
49
13
4

 

 

-
16
T
H
E
-
A
T
O
M
I
C
-
N
U
C
L
E
U
S
-
-
-
-
-
--
-
-
-
-
-
-
-
8
-
-
-
-
6
-
9
-
-
5
-
-
-
-
-
1
+
=
29
2+9
=
11
1+1
2
-
2
-
-
-
8
-
-
-
-
15
-
9
-
-
14
-
-
-
-
-
19
+
=
65
6+5
=
11
1+1
2
-
2
-
16
T
H
E
-
A
T
O
M
I
C
-
N
U
C
L
E
U
S
-
-
-
-
-
--
-
-
-
-
-
-
2
-
5
-
1
2
-
4
-
3
-
-
3
3
3
5
3
-
+
=
34
3+4
=
7
-
7
-
7
-
-
20
-
5
-
1
20
-
13
-
3
-
-
21
3
12
5
21
-
+
=
124
1+2+4
=
7
-
7
-
7
-
16
T
H
E
-
A
T
O
M
I
C
-
N
U
C
L
E
U
S
-
-
-
-
-
--
-
-
-
-
-
-
20
8
5
-
1
20
15
13
9
3
-
14
21
3
12
5
21
19
+
=
189
1+8+9
=
18
1+8
9
-
9
-
-
2
8
5
-
1
2
6
4
9
3
-
5
3
3
3
5
3
1
+
=
63
6+3
=
9
-
9
-
9
-
16
T
H
E
-
A
T
O
M
I
C
-
N
U
C
L
E
U
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
2
=
2
-
2
-
``-
2
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
2
=
4
-
4
-
-
-
-
-
-
-
-
-
-
-
3
-
-
3
3
3
-
3
-
-
-
3
occurs
x
5
=
15
1+5
6
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
-
-
-
-
4
occurs
x
1
=
4
-
4
-
-
-
-
5
-
-
-
-
-
-
-
-
5
-
-
-
5
-
-
-
-
5
occurs
x
3
=
15
1+5
6
-
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
6
occurs
x
1
=
6
-
6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
-
8
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
-
9
7
16
T
H
E
-
A
T
O
M
I
C
-
N
U
C
L
E
U
S
-
-
38
-
-
16
-
63
-
45
-
1+6
-
-
-
-
-
-
-
-
-
3
-
-
3
3
3
-
3
-
-
-
3+8
-
-
1+6
-
6+3
-
4+5
7
7
T
H
E
-
A
T
O
M
I
C
-
N
U
C
L
E
U
S
-
-
11
-
-
7
-
9
-
9
-
-
2
8
5
-
1
2
6
3
9
3
-
5
3
3
3
5
3
1
-
-
1+1
-
-
-
-
-
-
-
7
7
T
H
E
-
A
T
O
M
I
C
-
N
U
C
L
E
U
S
-
-
2
-
-
7
-
9
-
9

 

 

-
19
D
A
R
K
-
M
A
T
T
E
R
-
M
I
N
D
-
L
I
G
H
T
-
-
-
-
-
--
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
5
-
-
-
9
-
8
-
+
=
31
3+1
=
4
=
4
=
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
14
-
-
-
9
-
8
-
+
=
40
4+0
=
4
=
4
=
4
-
19
D
A
R
K
-
M
A
T
T
E
R
-
M
I
N
D
-
L
I
G
H
T
-
-
-
-
-
--
-
-
-
-
-
-
4
1
9
2
-
4
1
2
2
5
9
-
4
-
-
4
-
3
-
7
-
2
+
=
59
5+9
=
14
1+4
5
=
5
-
-
4
1
18
11
-
13
1
20
20
5
18
-
13
-
-
4
-
12
-
7
-
20
+
=
167
1+6+7
=
14
1+4
5
=
5
-
19
D
A
R
K
-
M
A
T
T
E
R
-
M
I
N
D
-
L
I
G
H
T
-
-
-
-
-
--
-
-
-
-
-
-
4
1
18
11
-
13
1
20
20
5
18
-
13
9
14
4
-
12
9
7
8
20
+
=
207
2+0+7
=
9
=
9
=
9
-
-
4
1
9
2
-
4
1
2
2
5
9
-
4
9
5
4
-
3
9
7
8
2
+
=
90
9+0
=
9
=
9
=
9
-
19
D
A
R
K
-
M
A
T
T
E
R
-
M
I
N
D
-
L
I
G
H
T
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
-
-
-
-
2
-
-
-
2
2
-
-
-
-
-
-
-
-
-
-
-
-
2
-
-
2
occurs
x
4
=
8
=
8
-
-
-
-
-
-
-
--
--
--
--
--
--
--
-
--
--
--
--
3
-
-
-
-
-
-
3
occurs
x
1
=
3
=
3
-
-
4
-
-
-
-
4
-
-
-
-
-
-
4
-
--
4
-
-
-
-
-
-
-
-
4
occurs
x
4
=
16
1+6
7
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
-
5
-
-
-
-
-
-
-
-
-
5
occurs
x
2
=
10
1+0
1
6
-
-
-
-
-
--
--
--
--
--
--
--
--
-
--
--
--
--
-
-
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
-
-
-
--
--
--
--
--
--
--
-
--
--
--
--
-
-
7
-
-
-
-
7
occurs
x
1
=
7
=
7
-
-
-
-
-
-
-
--
--
--
--
--
--
--
-
--
--
--
--
-
-
-
8
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
9
-
-
-
-
-
-
-
9
-
-
9
-
-
-
-
9
-
-
-
-
-
9
occurs
x
4
=
36
3+6
9
6
19
D
A
R
K
-
M
A
T
T
E
R
-
M
I
N
D
-
L
I
G
H
T
-
-
39
-
-
19
-
90
-
45
-
1+9
-
-
9
-
-
-
-
-
-
-
9
-
-
9
-
-
-
-
9
-
-
-
-
-
3+9
-
-
1+9
-
9+0
-
4+5
6
10
D
A
R
K
-
M
A
T
T
E
R
-
M
I
N
D
-
L
I
G
H
T
-
-
12
-
-
10
-
9
-
9
-
1+0
4
1
9
2
-
4
1
2
2
5
9
-
4
9
5
4
-
3
9
7
8
2
-
-
1+2
-
-
1+0
-
9+0
-
4+5
6
1
D
A
R
K
-
M
A
T
T
E
R
-
M
I
N
D
-
L
I
G
H
T
-
-
3
-
-
1
-
9
-
9

 

 

BEWARE NOT THIS GREEK AND THE GIFTS HE IS CARRYING

 

 

2061

ODYSSEY THREE

Arthur C. Clarke 1987

Page 13 (number omitted)

"THE MAGIC MOUNTAIN"

 

 

THE MAGIC MOUNTAIN

Thomas Mann 1924

Page 706

THE

THUNDERBOLT

 

WHY SMASH ATOMS

A. K. Solomon 1940

Page 77

"Once the fairy tale hero has penetrated the ring of fire round the magic mountain he is free to woo the heroine in her castle on the mountain top."

 

 

THE FAIRY RING

THE FAERIE RING

THE FIERY RING

THE RING OF FIRE

RINGING THE FIRE

RINGING THE CHANGES

I DELIGHT IN LIGHT IN LIGHT I DELIGHT

THE LIGHT IS RISING RISING IS THE LIGHT

 

 

2
IS
28
10
1
9
UNIVERSAL
121
40
4
4
MIND
40
22
4
3
THE
33
15
6
4
MIND
40
22
4
2
OF
21
12
3
9
HUMANKIND
95
41
5
33
First Total
378
162
27
3+3
Add to Reduce
3+7+8
1+6+2
2+7
6
Second Total
18
9
9
-
Reduce to Deduce
1+8
-
-
6
Essence of Number
9
9
9

 

 

9
UNIVERSAL
121
40
4
4
MIND
40
22
4
2
IS
28
10
1
3
THE
33
15
6
4
MIND
40
22
4
2
OF
21
12
3
9
HUMANKIND
95
41
5
33
First Total
378
162
27
3+3
Add to Reduce
3+7+8
1+6+2
2+7
6
Second Total
18
9
9
-
Reduce to Deduce
1+8
-
-
6
Essence of Number
9
9
9

 

 

T
=
2
3
THE
33
15
6
E
=
5
N
=
5
6
NINETY
87
33
6
Y
=
7
N
=
5
4
NINE
42
24
6
E
=
5
N
=
5
5
NAMES
52
16
7
S
=
1
O
=
6
2
OF
21
12
3
F
=
6
G
=
7
3
GOD
26
17
8
D
=
4
-
-
30
23
Add to Reduce
261
117
36
-
-
28
-
-
3+0
2+3
Reduce to Deduce
2+6+1
1+1+7
3+6
-
-
10
-
-
3
5
Essence of Number
9
9
9
-
-
1

 

 

-
23
T
H
E
-
N
I
N
E
T
Y
-
N
I
N
E
-
N
A
M
E
S
-
O
F
-
G
O
D
-
-
-
-
-
--
-
-
-
-
-
-
-
5
-
-
5
9
5
-
-
-
-
5
9
5
-
-
5
-
-
-
1
-
6
-
-
-
6
-
+
=
64
6+0
=
10
1+0
1
=
1
-
-
-
8
-
-
14
9
14
-
-
-
-
14
9
14
-
-
14
-
-
-
19
-
15
-
-
-
15
-
+
=
145
1+4+5
=
10
1+0
1
=
1
-
23
T
H
E
-
N
I
N
E
T
Y
-
N
I
N
E
-
N
A
M
E
S
-
O
F
-
G
O
D
-
-
-
-
-
--
-
-
-
-
-
-
2
-
5
-
-
-
-
5
2
7
-
-
-
-
5
-
-
1
4
5
-
-
-
6
-
7
-
4
+
=
53
5+3
=
8
=
8
=
8
-
-
20
-
5
-
-
-
-
5
20
25
-
-
-
-
5
-
-
1
13
5
-
-
-
6
-
7
-
4
+
=
116
1+1+6
=
8
=
8
=
8
-
23
T
H
E
-
N
I
N
E
T
Y
-
N
I
N
E
-
N
A
M
E
S
-
O
F
-
G
O
D
-
-
-
-
-
--
-
-
-
-
-
-
20
8
5
-
14
9
14
5
20
25
-
14
9
14
5
-
14
1
13
5
19
-
15
6
-
7
15
4
+
=
261
2+6+1
=
9
=
9
=
9
-
-
2
8
5
-
5
9
5
5
2
7
-
5
9
5
5
-
5
1
4
5
1
-
6
6
-
7
6
4
+
=
117
1+1+7
=
9
=
9
=
9
-
23
T
H
E
-
N
I
N
E
T
Y
-
N
I
N
E
-
N
A
M
E
S
-
O
F
-
G
O
D
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
-
-
-
-
-
-
-
1
occurs
x
2
=
2
=
2
-
``-
2
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
-
4
-
-
4
occurs
x
2
=
8
=
8
-
-
-
-
5
-
5
-
5
5
-
-
-
5
-
5
5
-
5
-
-
5
-
-
-
-
-
-
-
-
-
-
5
occurs
x
9
=
45
4+5
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
6
-
-
6
-
-
-
6
occurs
x
3
=
18
1+8
9
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
7
occurs
x
2
=
14
1+4
5
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
-
-
-
9
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
occurs
x
2
=
18
1+8
9
3
23
T
H
E
-
N
I
N
E
T
Y
-
N
I
N
E
-
N
A
M
E
S
-
O
F
-
G
O
D
-
-
42
-
-
23
-
117
-
54
-
2+3
-
-
5
-
5
-
5
5
-
-
-
5
-
5
5
-
5
-
-
5
-
-
-
-
-
-
-
-
-
-
4+2
-
-
2+3
-
1+1+7
-
5+4
3
5
T
H
E
-
N
I
N
E
T
Y
-
N
I
N
E
-
N
A
M
E
S
-
O
F
-
G
O
D
-
-
4
-
-
5
-
9
-
9
-
-
2
8
5
-
5
9
5
5
2
7
-
5
9
5
5
-
5
1
4
5
1
-
6
6
-
7
6
4
-
-
-
-
-
-
-
-
-
-
3
5
T
H
E
-
N
I
N
E
T
Y
-
N
I
N
E
-
N
A
M
E
S
-
O
F
-
G
O
D
-
-
4
-
-
5
-
9
-
9

 

 

-
20
T
H
E
-
G
R
E
A
T
E
S
T
-
N
A
M
E
-
O
F
-
G
O
D
-
-
-
-
-
--
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
1
-
-
5
-
-
-
-
6
-
-
-
6
-
+
=
26
2+6
=
8
=
8
=
8
-
-
-
8
-
-
-
-
-
-
-
-
19
-
-
14
-
-
-
-
15
-
-
-
15
-
+
=
71
7+1
=
8
=
8
=
8
-
20
T
H
E
-
G
R
E
A
T
E
S
T
-
N
A
M
E
-
O
F
-
G
O
D
-
-
-
-
-
--
-
-
-
-
-
-
2
-
5
-
7
9
5
1
2
5
-
2
-
-
1
4
5
-
-
6
-
7
-
4
+
=
65
6+5
=
11
1+1
2
=
2
-
-
20
-
5
-
7
18
5
1
20
5
-
20
-
-
1
13
5
-
-
6
-
7
-
4
+
=
137
1+3+7
=
11
1+1
2
=
2
-
20
T
H
E
-
G
R
E
A
T
E
S
T
-
N
A
M
E
-
O
F
-
G
O
D
-
-
-
-
-
--
-
-
-
-
-
-
20
8
5
-
7
18
5
1
20
5
19
20
-
14
1
13
5
-
15
6
-
7
15
4
+
=
208
2+0+8
=
10
1+0
1
=
1
-
-
2
8
5
-
7
9
5
1
2
5
1
2
-
5
1
4
5
-
6
6
-
7
6
4
+
=
91
9+1
=
10
1+0
1
=
1
-
20
T
H
E
-
G
R
E
A
T
E
S
T
-
N
A
M
E
-
O
F
-
G
O
D
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
-
-
-
1
-
-
-
-
-
-
-
-
-
-
-
1
occurs
x
3
=
3
=
3
-
``-
2
-
-
-
-
-
-
-
2
-
-
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
occurs
x
3
=
6
=
6
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
-
-
-
-
-
4
-
-
4
occurs
x
2
=
8
=
8
-
-
-
-
5
-
-
-
5
-
-
5
-
-
-
5
-
-
5
-
-
-
-
-
-
-
-
-
5
occurs
x
5
=
25
2+5
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
6
-
-
6
-
-
-
6
occurs
x
3
=
18
1+8
9
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
7
occurs
x
2
=
14
1+4
5
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
occurs
x
1
=
8
=
8
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
3
20
T
H
E
-
G
R
E
A
T
E
S
T
-
N
A
M
E
-
O
F
-
G
O
D
-
-
42
-
-
20
-
91
-
55
-
2+0
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4+2
-
-
2+0
-
9+1
``-
5+5
3
2
T
H
E
-
G
R
E
A
T
E
S
T
-
N
A
M
E
-
O
F
-
G
O
D
-
-
4
-
-
2
-
10
-
10
-
-
2
8
5
-
7
9
5
1
2
5
1
2
-
5
1
4
5
-
6
6
-
7
6
4
-
-
-
-
-
-
-
1+0
-
1+0
3
2
T
H
E
-
G
R
E
A
T
E
S
T
-
N
A
M
E
-
O
F
-
G
O
D
-
-
4
-
-
2
-
1
-
1

 
 
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