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Evokation
  
 
Index
  

 

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

 

 

ADVENT 749 ADVENT

 

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

 

 

METRIC TENSOR
SEDNA, ANDES, CUZCO, INCA, INCAS
SHAMAN, SHAMANS
LLULLAILLACO, LLU, LA, I, LLA, CO
LLULLAILLACO, L,L,U,L,L,A,I,L,L,A,C,O
LLULLAILLACO
CAPACOCHA, C, APA, C, O, C, HA
CAPOCOCHA, C, APA, CO, C, HA
CAPACOCHA, C, APA, CO, CHA
CAPACOCHA, CAPA, COCHA
I, NCA, INCA
THE, SON, OF, THE, SUN
MOUNT, AMPATO
ANDES, AND, DIES
THE SON OF THE SUN
CHRIST, THE SON OF THE SUN
MONEY
GENERATION, REGENERATION
ANNUAL
CALCHAQUIES
INDIGENOUS
INDIGENOUS, I, ND, I, GEN, OU, S
TRY
IDEAS
IDEA
CALCULI

 

 

-
12
M
E
T
R
I
C
-
T
E
N
S
O
R
-
-
-
-
-
=
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
5
1
6
-
+
=
21
2+1
=
3
-
3
=
3
-
-
-
-
-
-
9
-
-
-
-
14
19
15
-
+
=
57
5+7
=
12
1+2
3
=
3
-
12
M
E
T
R
I
C
-
T
E
N
S
O
R
-
-
-
-
-
-
-
-
-
-
-
-
4
5
2
9
-
3
-
2
5
-
-
-
9
+
=
39
3+9
=
12
1+2
3
=
3
-
-
13
5
20
18
-
3
-
20
5
-
-
-
18
+
=
102
1+0+2
=
3
-
3
=
3
-
12
M
E
T
R
I
C
-
T
E
N
S
O
R
-
-
-
-
-
-
-
-
-
-
-
-
13
5
20
18
9
3
-
20
5
14
19
15
18
+
=
159
1+5+9
=
15
1+5
6
=
6
-
-
4
5
2
9
9
3
-
2
5
5
1
6
9
+
=
60
6+0
=
6
1+6
6
=
6
-
12
M
E
T
R
I
C
-
T
E
N
S
O
R
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
-
2
-
-
-
-
2
-
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
-
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
=
6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
-
-
-
-
-
-
-
-
-
-
-
9
9
-
-
-
-
-
-
-
9
-
-
9
occurs
x
3
=
27
2+7
9
15
12
M
E
T
R
I
C
-
T
E
N
S
O
R
-
-
30
-
-
12
-
60
-
33
1+5
1+2
-
-
-
9
9
-
-
-
-
-
-
-
9
-
-
3+0
-
-
1+2
-
6+0
-
3+3
6
3
M
E
T
R
I
C
-
T
E
N
S
O
R
-
-
3
-
-
3
-
6
-
6
-
-
4
5
2
9
9
3
-
2
5
5
1
6
9
-
-
-
-
-
-
-
-
-
-
6
3
M
E
T
R
I
C
-
T
E
N
S
O
R
-
-
3
-
-
6
-
7
-
7

 

 

12
M
E
T
R
I
C
-
T
E
N
S
O
R
-
-
-
-
-
=
-
-
-
-
-
-
-
-
-
9
-
-
-
-
5
1
6
-
+
=
21
2+1
=
3
-
3
=
3
-
-
-
-
-
9
-
-
-
-
14
19
15
-
+
=
57
5+7
=
12
1+2
3
=
3
12
M
E
T
R
I
C
-
T
E
N
S
O
R
-
-
-
-
-
-
-
-
-
-
-
4
5
2
9
-
3
-
2
5
-
-
-
9
+
=
39
3+9
=
12
1+2
3
=
3
-
13
5
20
18
-
3
-
20
5
-
-
-
18
+
=
102
1+0+2
=
3
-
3
=
3
12
M
E
T
R
I
C
-
T
E
N
S
O
R
-
-
-
-
-
-
-
-
-
-
-
13
5
20
18
9
3
-
20
5
14
19
15
18
+
=
159
1+5+9
=
15
1+5
6
=
6
-
4
5
2
9
9
3
-
2
5
5
1
6
9
+
=
60
6+0
=
6
1+6
6
=
6
12
M
E
T
R
I
C
-
T
E
N
S
O
R
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
2
-
-
-
-
2
-
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
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
=
6
-
-
-
-
9
9
-
-
-
-
-
-
-
9
-
-
9
occurs
x
3
=
27
2+7
9
12
M
E
T
R
I
C
-
T
E
N
S
O
R
-
-
30
-
-
12
-
60
-
33
1+2
-
-
-
9
9
-
-
-
-
-
-
-
9
-
-
3+0
-
-
1+2
-
6+0
-
3+3
3
M
E
T
R
I
C
-
T
E
N
S
O
R
-
-
3
-
-
3
-
6
-
6
-
4
5
2
9
9
3
-
2
5
5
1
6
9
-
-
-
-
-
-
-
-
-
-
3
M
E
T
R
I
C
-
T
E
N
S
O
R
-
-
3
-
-
6
-
7
-
7

 

 

12
M
E
T
R
I
C
T
E
N
S
O
R
-
-
-
-
-
=
-
-
-
-
-
-
-
-
-
9
-
-
-
5
1
6
-
+
=
21
2+1
=
3
-
3
=
3
-
-
-
-
-
9
-
-
-
14
19
15
-
+
=
57
5+7
=
12
1+2
3
=
3
12
M
E
T
R
I
C
T
E
N
S
O
R
-
-
-
-
-
-
-
-
-
-
-
4
5
2
9
-
3
2
5
-
-
-
9
+
=
39
3+9
=
12
1+2
3
=
3
-
13
5
20
18
-
3
20
5
-
-
-
18
+
=
102
1+0+2
=
3
-
3
=
3
12
M
E
T
R
I
C
T
E
N
S
O
R
-
-
-
-
-
-
-
-
-
-
-
13
5
20
18
9
3
20
5
14
19
15
18
+
=
159
1+5+9
=
15
1+5
6
=
6
-
4
5
2
9
9
3
2
5
5
1
6
9
+
=
60
6+0
=
6
1+6
6
=
6
12
M
E
T
R
I
C
T
E
N
S
O
R
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
1
occurs
x
1
=
1
=
1
-
-
-
2
-
-
-
2
-
-
-
-
-
-
-
2
occurs
x
2
=
4
=
4
-
-
-
-
-
-
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
=
6
-
-
-
-
9
9
-
-
-
-
-
-
9
-
-
9
occurs
x
3
=
27
2+7
9
12
M
E
T
R
I
C
T
E
N
S
O
R
-
-
30
-
-
12
-
60
-
33
1+2
-
-
-
9
9
-
-
-
-
-
-
9
-
-
3+0
-
-
1+2
-
6+0
-
3+3
3
M
E
T
R
I
C
T
E
N
S
O
R
-
-
3
-
-
3
-
6
-
6
-
4
5
2
9
9
3
2
5
5
1
6
9
-
-
-
-
-
-
-
-
-
-
3
M
E
T
R
I
C
T
E
N
S
O
R
-
-
3
-
-
6
-
7
-
7

 

 

S
=
1
5
SEDNA
43
16
7
A
=
1
5
ANDES
43
16
7
C
=
3
5
CUZCO
68
23
5
I
=
9
4
INCA
27
18
9
I
=
9
5
INCAS
46
19
1

 

 

S
=
1
6
SHAMAN
56
20
2
S
=
1
7
SHAMANS
75
21
3

 

 

IN

MEMORIAM

"SUFFER LITTLE CHILDREN TO COME UNTO ME"

 

 

-
-
-
-
-
-
-
-
LLULLAILLACO

-

-
-
L
3
-
-
-
-
L
1
L

12

3
3
L
3
-
-
-
-
L
1
L

12

3
3
U
3
-
-
-
-
U
1
U

21

3
3
L
3
-
-
-
-
L
1
L

12

3
3
L
3
-
-
-
-
L
1
L

12

3
3
-
-
A
-
-
-
A
1
A

1

1
1
-
-
-
I
-
-
I
1
I

9

9
9
L
3
-
-
-
-
L
1
L

12

3
3
L
3
-
-
-
-
L
1
L

12

3
3
-
-
A
-
-
-
A
1
A

1

1
1
C
3
-
-
-
-
C
1
C

3

3
3
-
-
-
-
O
-
O
1
O

15

6
6
-
24
-
-
-
-
-
12
LLULLAILLACO

122

41

41
-
2+4
-
-
-
-
-
1+2
-
1+2+2
4+1
4+1
-
6
-
-
-
-
-
3
LLULLAILLACO

5

5

5
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
3
LLULLAILLACO

5

5

5

 

 

Children of Llullaillaco, sacrificed by the Incas 500 years ago.

It is believed the Children of Llullaillaco, as they have come to be known, were sacrificed during a ceremony thanking the Inca gods for the annual corn ... www.democraticunderground.com/discuss/ duboard.

http://news.bbc.co.uk/1/hi/world/americas/6983300.stm

Mummified Inca maiden wows crowds

"A mummy of an Inca girl, described as "perfect" by the archaeologists who found her in 1999, has gone on display for the first time in Argentina .

Hundreds of people crowded into a museum in the north-western city of Salta to see "la Doncella", the Maiden.

The remains of the girl, who was 15 when she died, were found in an icy pit on top of a volcano in the Andes, along with a younger boy and girl. Researchers believe they were sacrificed by the Incas 500 years ago. The three were discovered at a height of 6,700m (22,000ft) on Mount Llullaillaco, a volcano in north-west Argentina on the border with Chile. At the time, the archaeologist leading the team, Dr Johan Reinhard, said they appeared "the best preserved of any mummy I've seen". It is believed the Children of Llullaillaco, as they have come to be known, were sacrificed during a ceremony thanking the Inca gods for the annual corn harvest.

'Great mistake'

The mummy of la Doncella is on display in a chamber that is filled with cold air that recreates the sub-freezing conditions in which she was found. Visitors told Argentina media they were impressed at the mummy's state of conservation. "I'm amazed," one woman said. "You just expect her at any moment to get up and start talking." But the exhibition has angered several indigenous groups who campaigned to stop the mummy from going on display.

Miguel Suarez from the Calchaquies valley tribes in and around Salta told the Associated Press news agency that the exhibit was "a great mistake", adding that he hoped visitors would show respect for the dead."

 

 

-
LLULLAILLACO

-

-
-
3
L+L+U

45

9
9
2
L+L+A

25

7
7
1
I

9

9
9
3
L+L+A

25

7
7
2
C+O

18

9
9
12
LLULLAILLACO

122

41

41
1+2
-
1+2+2
4+1
4+1
3
LLULLAILLACO

5

5

5
-
-
-
-
-
3
LLULLAILLACO

5

5

5

 

 

-
-
-
-
-
-
-
-
LLULLAILLACO

-

-
-
L
3
-
-
-
-
L
1
L

12

3
3
L
3
-
-
-
-
L
1
L

12

3
3
U
3
-
-
-
-
U
1
U

21

3
3
L
3
-
-
-
-
L
1
L

12

3
3
L
3
-
-
-
-
L
1
L

12

3
3
-
-
A
-
-
-
A
1
A

1

1
1
-
-
-
I
-
-
I
1
I

9

9
9
L
3
-
-
-
-
L
1
L

12

3
3
L
3
-
-
-
-
L
1
L

12

3
3
-
-
A
-
-
-
A
1
A

1

1
1
C
3
-
-
-
-
C
1
C

3

3
3
-
-
-
-
O
-
O
1
O

15

6
6
-
24
-
-
-
-
-
12
LLULLAILLACO

122

41

41
-
2+4
-
-
-
-
-
1+2
-
1+2+2
4+1
4+1
-
6
-
-
-
-
-
3
LLULLAILLACO

5

5

5
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
-
-
-
3
LLULLAILLACO

5

5

5

 

The remains of the girl, who was 15 when she died, were found in an icy pit on top of a volcano in the Andes, along with a younger boy and girl. Researchers believe they were sacrificed by the Incas 500 years ago. The three were discovered at a height of 6,700m (22,000ft) on Mount Llullaillaco, a volcano in north-west Argentina on the border with Chile.

 

-
-
-
-
-
LLULLAILLACO

-

-
-
-
-
L
L
3
-
1
L

12

3
3
-
3
L
L
3
-
1
L

12

3
3
-
3
U
U
3
-
1
U

21

3
3
-
3
L
L
3
-
1
L

12

3
3
-
3
L
L
3
-
1
L

12

3
3
-
3
A
-
-
-
1
A

1

1
1
-
-
I
-
-
-
1
I

9

9
9
-
-
L
L
3
-
1
L

12

3
3
-
3
L
L
3
-
1
L

12

3
3
-
3
A
-
-
-
1
A

1

1
1
-
-
C
C
3
-
1
C

3

3
3
-
3
O
-
-
-
1
O

15

6
6
-
-
-
-
24
-
12
LLULLAILLACO

122

41

41
-

41
-
-
2+4
-
1+2
-
1+2+2
4+1
4+1
-
4+1
-
-
6
-
3
LLULLAILLACO

5

5

5
-

5
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
3
LLULLAILLACO

5

5

5
-

5

 

 

-
12
L
L
U
L
L
A
I
L
L
A
C
O
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
6
+
=
15
1+5
=
6
=
6
=
6
-
-
-
-
-
-
-
-
9
-
-
-
-
15
+
=
24
2+4
=
6
=
6
=
6
-
12
L
L
U
L
L
A
I
L
L
A
C
O
-
-
-
-
-
-
-
-
-
-
-
-
3
3
3
3
3
1
-
3
3
1
3
-
+
=
26
2+6
=
8
=
8
=
8
-
-
12
12
21
12
12
1
-
12
12
1
3
-
+
=
98
9+8
=
17
1+7
8
=
8
-
12
L
L
U
L
L
A
I
L
L
A
C
O
-
-
-
-
-
-
-
-
-
-
-
-
12
12
21
12
12
1
9
12
12
1
3
15
+
=
122
1+2+2
=
5
=
5
=
5
-
-
3
3
3
3
3
1
9
3
3
1
3
6
+
=
41
4+1
=
5
=
5
=
5
-
12
L
L
U
L
L
A
I
L
L
A
C
O
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
-
-
-
-
1
occurs
x
2
=
2
=
2
2
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
2
TWO
2
-
-
-
-
-
-
-
3
3
3
3
3
-
-
3
3
-
3
--
-
-
3
occurs
x
8
=
24
2+4
6
4
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
5
FIVE
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
6
occurs
x
1
=
6
=
6
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
26
12
L
L
U
L
L
A
I
L
L
A
C
O
-
-
19
-
-
12
-
41
-
23
2+6
1+2
3
3
3
3
3
-
-
3
3
-
3
--
-
-
1+9
-
-
1+2
-
4+1
-
2+3
8
3
L
L
U
L
L
A
I
L
L
A
C
O
-
-
10
-
-
3
-
5
-
5
-
-
3
3
3
3
3
1
9
3
3
1
3
6
-
-
1+0
-
-
-
-
-
-
-
8
3
L
L
U
L
L
A
I
L
L
A
C
O
-
-
1
-
-
3
-
5
-
5

 

 

12
L
L
U
L
L
A
I
L
L
A
C
O
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
6
+
=
15
1+5
=
6
=
6
=
6
-
-
-
-
-
-
-
9
-
-
-
-
15
+
=
24
2+4
=
6
=
6
=
6
12
L
L
U
L
L
A
I
L
L
A
C
O
-
-
-
-
-
-
-
-
-
-
-
3
3
3
3
3
1
-
3
3
1
3
-
+
=
26
2+6
=
8
=
8
=
8
-
12
12
21
12
12
1
-
12
12
1
3
-
+
=
98
9+8
=
17
1+7
8
=
8
12
L
L
U
L
L
A
I
L
L
A
C
O
-
-
-
-
-
-
-
-
-
-
-
12
12
21
12
12
1
9
12
12
1
3
15
+
=
122
1+2+2
=
5
=
5
=
5
-
3
3
3
3
3
1
9
3
3
1
3
6
+
=
41
4+1
=
5
=
5
=
5
12
L
L
U
L
L
A
I
L
L
A
C
O
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
1
-
-
-
-
1
occurs
x
2
=
2
=
2
-
3
3
3
3
3
-
-
3
3
-
3
--
-
-
3
occurs
x
8
=
24
2+4
6
-
-
-
-
-
-
-
-
-
-
-
-
6
-
-
6
occurs
x
1
=
6
=
6
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
9
occurs
x
1
=
9
=
9
12
L
L
U
L
L
A
I
L
L
A
C
O
-
-
19
-
-
12
-
41
-
23
1+2
3
3
3
3
3
-
-
3
3
-
3
--
-
-
1+9
-
-
1+2
-
4+1
-
2+3
3
L
L
U
L
L
A
I
L
L
A
C
O
-
-
10
-
-
3
-
5
-
5
-
3
3
3
3
3
1
9
3
3
1
3
6
-
-
1+0
-
-
-
-
-
-
-
3
L
L
U
L
L
A
I
L
L
A
C
O
-
-
1
-
-
3
-
5
-
5

 

LOOK AT THE THREES LOOK AT THE THREES LOOK AT THE THREES THE THREES THE THREES

 

O
=
6
-
-
THREES
-
-
-
-
-
-
-
1
T
20
2
-
-
-
-
-
1
H
8
8
8
-
-
-
-
1
R
18
9
9
-
-
-
-
1
E
5
5
5
-
-
-
-
1
E
5
5
5
-
-
-
-
2
S
19
10
1
O
=
6
Q
6
THREES
75
39
30
-
-
-
-
-
-
7+4
3+9
3+0
O
=
6
Q
6
THREES
12
12
3
-
-
-
-
-
-
1+2
1+2
-
O
=
6
Q
6
THREES
3
3
3

 

LOOK AT THE THREES LOOK AT THE THREES LOOK AT THE THREES THE THREES THE THREES

 

8
CALCULUS
-
-
-
8
-
-
-
C
3
3
3
-
-
3
-
A
1
1
1
-
1
-
-
L
12
3
3
-
-
3
-
C
3
3
3
-
-
3
-
U
21
3
3
-
-
3
-
L
12
3
3
-
-
3
-
U
21
3
3
-
-
3
-
S
19
10
10
-
1
-
8
CALCULUS
92
29
29
-
2
18
-
-
9+2
2+9
2+9
-
-
1+8
8
CALCULUS
11
11
11
-
2
9
-
-
1+1
1+1
1+1
-
-
-
8
CALCULUS
2
2
2
-
2
9

 

 

T
=
2
-
3
THE
31
15
6
R
=
9
-
7
REALITY
90
36
9
W
=
5
-
6
WITHIN
83
38
2
T
=
2
-
3
THE
33
15
6
P
=
7
-
7
PATTERN
94
31
4
-
-
25
Q
26
Add to Reduce
333
135
27
-
-
2+5
-
2+6
Reduce to Deduce
3+3+3
1+3+5
2+7
-
-
7
-
8
Essence of Number
9
9
9

 

 

LOOK AT THE THREES LOOK AT THE THREES LOOK AT THE THREES THE THREES THE THREES

 

 

The civilisation we discuss, which does not appear to have found a need to develop writing, is that of the Incas. The Inca empire which existed in 1532, ... www-history.mcs.st-andrews.ac.uk/ HistTopics/Inca_mathematics

History topic: Mathematics of the Incas

It is often thought that mathematics can only develop after a civilisation has developed some form of writing. Although not easy for us to understand today, many civilisations reached highly advanced states without ever developing written records. Now of course it is difficult for us to know much about such civilisations since there is no written record to be studied today. This article looks at the mathematical achievements of one such civilisation.

The civilisation we discuss, which does not appear to have found a need to develop writing, is that of the Incas. The Inca empire which existed in 1532, before the Spanish conquest, was vast. It spread over an area which stretched from what is now the northern border of Ecuador to Mendoza in west-central Argentina and to the Maule River in central Chile. The Inca people numbered around 12 million but they were from many different ethnic groups and spoke about 20 different languages. The civilisation had reached a high level of sophistication with a remarkable system of roads, agriculture, textile design, and administration. Of course even if writing is not required to achieve this level, counting and recording of numerical information is necessary. The Incas had developed a method of recording numerical information which did not require writing. It involved knots in strings called quipu.

The quipu was not a calculator, rather it was a storage device. Remember that the Incas had no written records and so the quipu played a major role in the administration of the Inca empire since it allowed numerical information to be kept. Let us first describe the basic quipu, with its positional number system, and then look at the ways that it was used in Inca society.

The quipu consists of strings which were knotted to represent numbers. A number was represented by knots in the string, using a positional base 10 representation. If the number 586 was to be recorded on the string then six touching knots were placed near the free end of the string, a space was left, then eight touching knots for the 10s, another space, and finally 5 touching knots for the 100s.

(Illustration omitted) 586 on a quipu.

For larger numbers more knot groups were used, one for each power of 10, in the same way as the digits of the number system we use here are occur in different positions to indicate the number of the corresponding power of 10 in that position.

Now it is not quite true that the same knots were used irrespective of the position as would be the case in a true positional system. There seems only one exception, namely the unit position, where different styles of knots were used from those in the other positions. In fact two different styles were used in the units position, one style if the unit were a 1 and a second style if the unit were greater than one. Both these styles differed from the standard knot used for all other positions. The system had a zero position, for this would be represented as no knots in that position. This meant that the spacing had to be highly regular so that zero positions would be clear.

There are many drawings and descriptions of quipus made by the Spanish invaders. Garcilaso de la Vega, whose mother was an Inca and whose father was Spanish, wrote (see for example [5]):-

According to their position, the knots signified units, tens, hundreds, thousands, ten thousands and, exceptionally, hundred thousands, and they are all well aligned on their different cords as the figures that an accountant sets down, column by column, in his ledger.

Now of course recording a number on a string would, in itself, not be that useful. A quipu had many strings and there had to be some way that the string carrying the record of a particular number could be identified. The primary way this was done was by the use of colour. Numbers were recorded on strings of a particular colour to identify what that number was recording. For example numbers of cattle might be recorded on green strings while numbers of sheep might be recorded on white strings. The colours each had several meanings, some of which were abstract ideas, some concrete as in the cattle and sheep example. White strings had the abstract meaning of "peace" while red strings had the abstract meaning of "war"

As well as the colour coding, another way of distinguishing the strings was to make some strings subsidiary ones, tied to the middle of a main string rather than being tied to the main horizontal cord.

(Illustration omitted) Quipu with subsidiary cords.

We quote Garcilaso de la Vega again [5]:-

The ordinary judges gave a monthly account of the sentences they imposed to their superiors, and they in turn reported to their immediate superiors, and so on finally to the Inca or those of his Supreme Council. The method of making these reports was by means of knots, made of various colours, where knots of such and such colours denote that such and such crimes had been punished. Smaller threads attached to thicker cords were of different colours to signify the precise nature of the punishment that had been inflicted.

It was not only judges who sent quipus to be kept in a central record. The Inca king appointed quipucamayocs, or keepers of the knots, to each town. Larger towns might have had up to thirty quipucamayocs who were essentially government statisticians, keeping official census records of the population, records of the produce of the town, its animals and weapons. This and other information was sent annually to the capital Cuzco. There was even an official delivery service to take to quipus to Cuzco which consisted of relay runners who passed the quipus on to the next runner at specially constructed staging posts. The terrain was extremely difficult yet the Incas had constructed roads to make the passing of information by quipus surprisingly rapid.

Much information on the quipus comes from a letter of the Peruvian Felipe Guaman Poma de Ayala to the King of Spain, written about eighty years after the Spanish conquest of the Incas. This remarkable letter contains 1179 pages and there are several drawings which show quipus. A fascinating aspect of one of these drawing is a picture of a counting board in the bottom left hand corner of one of them. This is called the yupana and is presumed to be the counting board of the Incas.

(Illustration omitted) This is what the yupana looked like.

Interpretations of how this counting board, or Peruvian abacus, might have been used have been given by several authors, see for example [9] and [11]. However some historians are less certain that this really is a Peruvian abacus. For example [2] in which the authors write:-

It is unclear from Poma's commentary whether it is his version of a device associated with Spanish activities analogous to those of the person depicted or whether he is implying its association with the Incas. In either case, his commentary makes interpretation of the configuration and the meaning of the unfilled and filled holes highly speculative.

It is a difficult task to gain further insights into the mathematical understanding of the Incas. The book [6] by Urton is interesting for it examines the concept of number as understood by the Inca people. As one might expect, their concept of number was a very concrete one, unlike our concept of number which is a highly abstract one (although this is not really understood by many people). The concrete way of conceiving numbers is illustrated by different words used when describing properties of numbers. One example given in [6] is that of even and odd numbers. Now the ideas of an even number, say, relies on having an abstract concept of number which is independent of the objects being counted. However, the Peruvian languages had different words which applied to different types of objects. For example separate words occur for the idea of [6]:-

... the two together that make a pair ...

... the one together with its mate ...

... two - in reference to one thing that is divided into two parts ...

... a pair of two separate things bound intimately together, such as two bulls yoked together for ploughing ... etc.

This is a fascinating topic and one which deserves much further research. One wonders whether the Incas applied their number system to solve mathematical problems. Was it merely for recording? If the yupana really was an abacus then it must have been used to solve problems and this prompts the intriguing question of what these problems were. A tantalising glimpse may be contained in the writings of the Spanish priest José de Acosta who lived among the Incas from 1571 to 1586. He writes in his book Historia Natural Moral de las Indias which was published in Madrid in 1596:-

To see them use another kind of calculator, with maize kernels, is a perfect joy. In order to carry out a very difficult computation for which an able computer would require pen and paper, these Indians make use of their kernels. They place one here, three somewhere else and eight, I know not where. They move one kernel here and there and the fact is that they are able to complete their computation without making the smallest mistake. as a matter of fact, they are better at practical arithmetic than we are with pen and ink. Whether this is not ingenious and whether these people are wild animals let those judge who will! What I consider ascertain is that in what they undertake to do they are superior to us.

What a pity that de Acosta did not have the mathematical skills to give a precise description which would have allowed us to understand this method of calculation by the Incas.

Article by: J J O'Connor and E F Robertson January 2001 MacTutor History of Mathematics

[http://www-history.mcs.st-andrews.ac.uk/HistTopics/Inca_mathematics.html]

 

 

I

INCA

THE

SON OF THE SUN

 

 

- CAPACOCHA

-

-

 -
1 C

3

3

 3
3 A+P+A

18

9

 9
1 C

3

3

 3
1 O

15

6

 6
1 C

3

3

 3
2 H+A

9

9

 9
9 CAPACOCHA

-

-

 -

 

 

- CAPACOCHA

-

-

 -
1 C

3

3

 3
3 A+P+A

18

9

 9
1 C+O

18

9

 9
1 C

3

3

 3
2 H+A

9

9

 9
9 CAPACOCHA

-

-

 -

 

 

- CAPACOCHA

-

-

 -
1 C

3

3

 3
3 A+P+A

18

9

 9
2 C+O

18

9

 9
3 C+H+A

12

3

 3
9 CAPACOCHA

51

24

 24
-
 - 5+1 2+4 2+4
9 CAPACOCHA

6

6

 6

 

 

- CAPACOCHA

-

-

 -
4 CAPA

21

12

 3
5 COCHA

30

30

 3
9 CAPACOCHA

51

42

 6
-
 - 5+1 4+2 -
9 CAPACOCHA

6

6

 6

 

 

-
-
-
-
INCA
-
-
-
I
=
9
1
I
9
9
9
I
=
9
3
N+C+A
18
9
9
-
-
-
-
INCA
-
-
-

 

 

3
THE
33
15
6
3
SON
48
12
3
2
OF
21
12
3
3
THE
33
15
6
3
SUN
54
9
9
14
First Total
189
63
27
1+4
Add to Reduce
1+8+9
6+3
2+7
5
Second Total
18
9
9
-
Reduce to Deduce
1+8
-
-
5
Essence of Number
9
9
9

 

 

5
MOUNT
83
20
2
6
AMPATO
66
21
3

 

 

5
ANDES
43
16
7
3
AND
19
10
1
4
DIES
37
19
1

 

 

-
14
T
H
E
-
S
O
N
-
O
F
-
T
H
E
-
S
U
N
-
-
-
-
-
--
-
-
-
-
-
-
-
8
-
-
1
6
5
-
6
-
-
-
8
-
-
1
-
5
+
=
40
4+0
=
4
-
4
-
4
-
-
-
8
-
-
19
15
14
-
15
-
-
-
8
-
-
19
-
14
+
=
112
1+1+2
=
4
-
4
-
4
-
14
T
H
E
-
S
O
N
-
O
F
-
T
H
E
-
S
U
N
-
-
-
-
-
--
-
-
-
-
-
-
2
-
5
-
-
-
5
-
-
6
-
2
-
5
-
-
3
5
+
=
23
2+3
=
5
-
5
=
5
-
-
20
-
5
-
-
-
14
-
-
6
-
20
-
5
-
-
21
14
+
=
77
7+7
=
14
1+4
5
-
5
-
14
T
H
E
-
S
O
N
-
O
F
-
T
H
E
-
S
U
N
-
-
-
-
-
--
-
-
-
-
-
-
20
8
5
-
19
15
14
-
15
6
-
20
8
5
-
19
21
14
+
=
189
1+8+9
=
18
1+8
9
-
9
-
-
2
8
5
-
1
6
5
-
6
6
-
2
8
5
-
1
3
5
+
=
63
6+3
=
9
-
9
-
9
-
14
T
H
E
-
S
O
N
-
O
F
-
T
H
E
-
S
U
N
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
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
-
-
-
-
-
-
-
-
-
-
-
5
-
-
-
5
-
-
-
-
-
-
5
-
-
-
5
-
-
5
occurs
x
4
=
20
2+0
2
-
-
-
-
-
--
-
6
-
--
6
6
--
-
-
-
--
-
-
-
-
-
6
occurs
x
3
=
18
1+8
9
7
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
8
occurs
x
2
=
16
1+6
7
9
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
20
14
T
H
E
-
S
O
N
-
O
F
-
T
H
E
-
S
U
N
-
-
25
-
-
14
-
63
-
27
2+0
1+4
2
8
5
-
1
6
5
-
6
6
-
2
8
5
-
1
3
5
-
-
2+5
-
-
1+4
-
6+3
-
2+7
2
5
T
H
E
-
S
O
N
-
O
F
-
T
H
E
-
S
U
N
-
-
7
-
-
5
-
9
-
9

 

 

6
CHRIST
77
32
5
14
THE SON OF THE SUN
189
63
9
20
First Total
266
95
14
2+0
Add to Reduce
2+6+6
9+5
1+4
2
Second Total
14
14
5
-
Reduce to Deduce
1+4
1+4
-
2
Essence of Number
5
5
5

 

 

-`
5
M
O
N
E
Y
-
-
-
-
-
-
-
-
-
-
-
-
-
6
5
-
-
+
=
11
1+1
=
2
-
2
-
2
-
-
-
15
14
-
-
+
=
29
2+9
=
11
1+1
2
-
2
-
5
M
O
N
E
Y
-
-
-
-
-
-
-
-
-
-
-
-
4
-
-
5
7
+
=
16
1+6
=
7
=
7
=
7
-
-
13
-
-
5
25
+
=
43
4+3
=
7
=
7
=
7
-
5
M
O
N
E
Y
-
-
-
-
-
-
-
-
-
-
-
--
13
15
14
5
25
+
=
72
7+2
=
9
=
9
=
9
-
-
4
6
5
5
7
+
=
27
2+7
=
9
=
9
=
9
-
5
M
O
N
E
Y
T
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
-
2
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
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
occurs
x
1
=
7
=
7
8
-
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
-
9
-
-
-
-
-
-
-
23
5
M
O
N
E
Y
-
-
22
-
-
5
-
27
-
18
2+3
-
4
6
5
5
7
-
-
2+2
-
-
-
-
2+7
-
1+8
5
5
M
O
N
E
Y
-
-
4
-
-
5
-
9
-
9

 

 

5
M
O
N
E
Y
-
-
-
-
-
-
-
-
-
-
-
-
6
5
-
-
+
=
11
1+1
=
2
-
2
-
2
`-
-
15
14
-
-
+
=
29
2+9
=
11
1+1
2
-
2
5
M
O
N
E
Y
-
-
-
-
-
-
-
-
-
-
-
4
-
-
5
7
+
=
16
1+6
=
7
=
7
=
7
`-
13
-
-
5
25
+
=
43
4+3
=
7
=
7
=
7
5
M
O
N
E
Y
-
-
-
-
-
-
-
-
-
-
`-
13
15
14
5
25
+
=
72
7+2
=
9
=
9
=
9
-
4
6
5
5
7
+
=
27
2+7
=
9
=
9
=
9
5
M
O
N
E
Y
T
-
-
-
-
-
-
-
-
-
--
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
occurs
x
1
=
7
=
7
5
M
O
N
E
Y
-
-
22
-
-
5
-
27
-
18
-
4
6
5
5
7
-
-
2+2
-
-
-
-
2+7
-
1+8
5
M
O
N
E
Y
-
-
4
-
-
5
-
9
-
9

 

 

10
GENERATION
108
54
9
12
REGENERATION
131
68
5

 

 

6
ANNUAL
63
18
9

 

 

11
CALCHAQUIES
99
45
9

 

 

10
INDIGENOUS
117
54
9

 

 

-
INDIGENOUS

-

-

 -
1
I

9

9

 9
2
N+D

18

9

 9
1
I

9

9

 9
3
G+E+N

26

17

 8
2
O+U

36

9

 9
1
S

19

10

 1
10
INDIGENOUS
117
54
9
1+0
-

1+1+7

5+4
-
1
INDIGENOUS

9

9
9

 

 

3
TRY
63
18
9

 

 

-
5
I
D
E
A
S
-
-
-
-
-
-
-
-
-
-
9
-
-
-
1
+
=
10
1+0
=
1
-
1
-
-
9
-
-
-
19
+
=
28
2+8
=
10
1+0
1
-
5
I
D
E
A
S
-
-
-
-
-
-
-
-
-
-
-
4
5
1
-
+
=
10
1+0
=
1
-
1
-
5
I
D
E
A
S
-
-
-
-
-
-
-
-
-
-
9
4
5
1
19
+
=
38
3+8
=
11
1+1
2
-
-
9
4
5
1
1
+
=
20
2+0
=
2
-
2
-
5
I
D
E
A
S
-
-
-
-
-
--
-
-
-
-
-
-
-
1
1
-
-
1
occurs
x
2
=
2
2
-
-
-
-
-
-
-
-
2
TWO
2
-
-
-
3
-
-
-
-
-
-
-
-
3
THREE
3
-
-
-
-
-
-
4
-
-
-
-
-
4
occurs
x
1
=
4
-
-
-
-
5
-
-
-
-
5
occurs
x
1
=
5
6
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
7
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
8
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
9
-
-
-
-
-
-
9
occurs
x
1
=
9
26
5
I
D
E
A
S
-
-
19
-
-
5
-
20
2+6
-
9
-
-
-
-
-
-
1+9
-
-
-
-
2+0
8
5
I
D
E
A
S
-
-
10
-
-
5
-
2
-
-
9
4
5
1
1
-
-
1+0
-
-
-
-
-
8
5
I
D
E
A
S
-
-
4
-
-
3
-
7

 

IDEAS ASIDE

 

5
I
D
E
A
S
-
-
-
-
-
-
-
-
-
9
-
-
-
1
+
=
10
1+0
=
1
-
1
-
9
-
-
-
19
+
=
28
2+8
=
10
1+0
1
5
I
D
E
A
S
-
-
-
-
-
-
-
-
-
-
4
5
1
-
+
=
10
1+0
=
1
-
1
5
I
D
E
A
S
-
-
-
-
-
-
-
-
-
9
4
5
1
19
+
=
38
3+8
=
11
1+1
2
-
9
4
5
1
1
+
=
20
2+0
=
2
-
2
5
I
D
E
A
S
-
-
-
-
-
-
-
-
-
-
-
-
1
1
-
-
1
occurs
x
2
=
2
-
-
4
-
-
-
-
-
4
occurs
x
1
=
4
-
-
-
5
-
-
-
-
5
occurs
x
1
=
5
-
9
-
-
-
-
-
-
9
occurs
x
1
=
9
5
I
D
E
A
S
-
-
19
-
-
5
-
20
-
9
-
-
-
-
-
-
1+9
-
-
-
-
2+0
5
I
D
E
A
S
-
-
10
-
-
5
-
2
-
9
4
5
1
1
-
-
1+0
-
-
-
-
-
5
I
D
E
A
S
-
-
4
-
-
3
-
7

 

 

-
4
I
D
E
A
-
-
-
-
-
-
-
-
-
-
9
-
-
-
+
=
9
-
=
9
-
9
-
4
I
D
E
A
-
-
-
-
-
-
-
-
-
-
-
4
5
1
+
=
10
1+0
=
1
-
1
-
4
I
D
E
A
-
-
-
-
-
-
-
-
-
-
9
4
5
1
+
=
19
1+9
=
10
1+0
1
-
4
I
D
E
A
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
1
=
1
2
-
-
-
-
-
-
-
2
-
-
-
-
-
3
-
-
-
-
-
-
-
3
-
-
-
-
-
-
-
-
4
-
-
-
-
4
occurs
x
1
=
4
-
-
-
-
5
-
-
-
5
occurs
x
1
=
5
6
-
-
-
-
-
-
-
6
-
-
-
-
-
7
-
-
-
-
-
-
-
7
-
-
-
-
-
8
-
-
-
-
-
-
-
8
-
-
-
-
-
-
-
9
-
-
-
-
-
9
occurs
x
1
=
9
26
4
I
D
E
A
-
-
19
-
-
4
-
19
2+6
-
9
-
-
-
-
-
1+9
-
-
-
-
1+9
8
4
I
D
E
A
-
-
10
-
-
4
-
10
-
-
9
4
5
1
-
-
1+0
-
-
-
-
1+0
8
4
I
D
E
A
-
-
1
-
-
4
-
1

 

 

4
I
D
E
A
-
-
-
-
-
-
-
-
-
9
-
-
-
+
=
9
-
=
9
-
9
4
I
D
E
A
-
-
-
-
-
-
-
-
-
-
4
5
1
+
=
10
1+0
=
1
-
1
4
I
D
E
A
-
-
-
-
-
-
-
-
-
9
4
5
1
+
=
19
1+9
=
10
1+0
1
4
I
D
E
A
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
1
occurs
x
1
=
1
-
-
4
-
-
-
-
4
occurs
x
1
=
4
-
-
-
5
-
-
-
5
occurs
x
1
=
5
-
9
-
-
-
-
-
9
occurs
x
1
=
9
4
I
D
E
A
-
-
19
-
-
4
-
19
-
9
-
-
-
-
-
1+9
-
-
-
-
1+9
4
I
D
E
A
-
-
10
-
-
4
-
10
-
9
4
5
1
-
-
1+0
-
-
-
-
1+0
4
I
D
E
A
-
-
1
-
-
4
-
1

 

 

11
7
C
A
L
C
U
L
I
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
+
=
9
-
=
9
-
9
=
9
-
-
-
-
-
-
-
-
9
+
=
9
-
=
9
-
9
=
9
-
7
C
A
L
C
U
L
I
-
-
-
-
-
-
-
-
-
-
-
-
3
1
3
3
3
3
-
+
=
16
1+6
=
7
-
7
=
7
-
-
3
1
12
3
21
12
-
+
=
52
5+2
=
7
-
7
=
7
11
7
C
A
L
C
U
L
I
-
-
-
-
-
-
-
-
-
-
-
-
3
1
12
3
21
12
9
+
=
61
6+1
=
7
-
7
=
7
-
-
3
1
3
3
3
3
9
+
=
25
2+5
=
7
-
7
=
7
11
7
C
A
L
C
U
L
I
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
2
-
-
-
-
-
-
-
-
-
-
2
TWO
-
-
-
-
-
-
-
-
3
-
3
3
3
3
-
-
-
3
occurs
x
5
=
15
1+5
6
4
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
5
FIVE
5
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
-
-
-
-
-
-
-
-
9
-
-
9
occurs
x
1
=
9
=
9
32
7
C
A
L
C
U
L
I
-
-
13
-
-
7
-
25
-
16
3+2
-
3
-
3
3
3
3
-
-
-
1+3
-
-
-
-
2+5
-
1+6
5
7
C
A
L
C
U
L
I
-
-
4
-
-
7
-
7
-
7
-
-
3
1
3
3
3
3
9
-
-
-
-
-
-
-
-
-
-
5
7
C
A
L
C
U
L
I
-
-
4
-
-
7
-
7
-
7

 

 

7
C
A
L
C
U
L
I
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
9
+
=
9
-
=
9
-
9
=
9
-
-
-
-
-
-
-
9
+
=
9
-
=
9
-
9
=
9
7
C
A
L
C
U
L
I
-
-
-
-
-
-
-
-
-
-
-
3
1
3
3
3
3
-
+
=
16
1+6
=
7
-
7
=
7
-
3
1
12
3
21
12
-
+
=
52
5+2
=
7
-
7
=
7
7
C
A
L
C
U
L
I
-
-
-
-
-
-
-
-
-
-
-
3
1
12
3
21
12
9
+
=
61
6+1
=
7
-
7
=
7
-
3
1
3
3
3
3
9
+
=
25
2+5
=
7
-
7
=
7
7
C
A
L
C
U
L
I
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
-
-
1
occurs
x
1
=
1
=
1
-
3
-
3
3
3
3
-
-
-
3
occurs
x
5
=
15
1+5
6
-
-
-
-
-
-
-
9
-
-
9
occurs
x
1
=
9
=
9
7
C
A
L
C
U
L
I
-
-
13
-
-
7
-
25
-
16
-
3
-
3
3
3
3
-
-
-
1+3
-
-
-
-
2+5
-
1+6
7
C
A
L
C
U
L
I
-
-
4
-
-
7
-
7
-
7
-
3
1
3
3
3
3
9
-
-
-
-
-
-
-
-
-
-
7
C
A
L
C
U
L
I
-
-
4
-
-
7
-
7
-
7

 

 

8
CALCULUS
-
-
-
8
-
-
-
C
3
3
3
-
-
3
-
A
1
1
1
-
1
-
-
L
12
3
3
-
-
3
-
C
3
3
3
-
-
3
-
U
21
3
3
-
-
3
-
L
12
3
3
-
-
3
-
U
21
3
3
-
-
3
-
S
19
10
10
-
1
-
8
CALCULUS
92
29
29
-
2
18
-
-
9+2
2+9
2+9
-
-
1+8
8
CALCULUS
11
11
11
-
2
9
-
-
1+1
1+1
1+1
-
-
-
8
CALCULUS
2
2
2
-
2
9

 

 

-
8
C
A
L
C
U
L
U
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
+
=
1
-
=
1
=
1
=
1
-
-
-
-
-
-
-
-
-
19
+
=
19
1+9
=
10
1+0
1
=
1
-
8
C
A
L
C
U
L
U
S
-
-
-
-
-
-
-
-
-
-
-
-
3
1
3
3
3
3
3
-
+
=
19
1+9
=
10
1+0
1
=
1
-
-
3
1
12
3
21
12
21
-
+
=
73
7+3
=
10
1+0
1
=
1
-
8
C
A
L
C
U
L
U
S
-
-
-
-
-
-
-
-
-
-
-
-
3
1
12
3
21
12
21
19
+
=
92
9+2
=
11
1+0
2
=
2
-
-
3
1
3
3
3
3
3
1
+
=
20
2+0
=
2
=
2
=
2
-
8
C
A
L
C
U
L
U
S
-
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
-
-
1
occurs
x
2
=
2
=
2
2
-
-
-
-
-
-
-
-
-
-
-
2
TWO
2
-
-
3
-
3
3
3
3
3
-
-
-
3
occurs
x
6
=
18
1+8
9
4
-
-
-
-
-
-
-
-
-
-
-
4
FOUR
4
-
-
-
-
-
5
-
-
-
-
-
-
-
-
-
-
-
5
FIVE
5
-
-
-
-
-
6
-
-
-
-
-
-
-
-
-
-
-
6
SIX
6
-
-
-
-
-
7
-
-
-
-
-
-
-
-
-
-
-
7
SEVEN
7
-
-
-
-
-
8
-
-
-
-
-
-
-
-
-
-
-
8
EIGHT
8
-
-
-
-
-
9
-
-
-
-
-
-
-
-
-
-
-
9
NINE
9
-
-
-
-
-
41
8
C
A
L
C
U
L
U
S
-
-
4
-
-
8
-
20
-
11
4+1
-
3
-
3
3
3
3
3
-
-
-
-
-
-
-
-
2+0
-
2+0
5
8
C
A
L
C
U
L
U
S
-
-
4
-
-
8
-
2
-
2
-
-
3
1
3
3
3
3
3
1
-
-
-
-
-
-
-
-
-
-
2
8
C
A
L
C
U
L
U
S
-
-
4
-
-
8
-
2
-
2

 

 

8
C
A
L
C
U
L
U
S
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
1
+
=
1
-
=
1
=
1
=
1
-
-
-
-
-
-
-
-
19
+
=
19
1+9
=
10
1+0
1
=
1
8
C
A
L
C
U
L
U
S
-
-
-
-
-
-
-
-
-
-
-
3
1
3
3
3
3
3
-
+
=
19
1+9
=
10
1+0
1
=
1
-
3
1
12
3
21
12
21
-
+
=
73
7+3
=
10
1+0
1
=
1
8
C
A
L
C
U
L
U
S
-
-
-
-
-
-
-
-
-
-
-
3
1
12
3
21
12
21
19
+
=
92
9+2
=
11
1+0
2
=
2
-
3
1
3
3
3
3
3
1
+
=
20
2+0
=
2
=
2
=
2
8
C
A
L
C
U
L
U
S
-
-
-
-
-
-
-
-
-
-
-
-
1
-
-
-
-
-
1
-
-
1
occurs
x
2
=
2
=
2
-
3
-
3
3
3
3
3
-
-
-
3
occurs
x
6
=
18
1+8
9
8
C
A
L
C
U
L
U
S
-
-
4
-
-
8
-
20
-
11
-
3
-
3
3
3
3
3
-
-
-
-
-
-
-
-
2+0
-
2+0
8
C
A
L
C
U
L
U
S
-
-
4
-
-
8
-
2
-
2
-
3
1
3
3
3
3
3
1
-
-
-
-
-
-
-
-
-
-
8
C
A
L
C
U
L
U
S
-
-
4
-
-
8
-
2
-
2

 
 
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