Dave

Many thanks for your e-mail of the 28th. Forgive me for not getting back earlier I have been very busy. Your 99 piece was interesting, especially the telephone number enigma which I solved over Christmas and which now follows. I tried to send it earlier but the format keeps getting lost. Given that you've sent another e-mail I thought I'd better reply to the first. I won't open your latest because it has an attachment. herewith the answer to the telephone enigma:

The Telephone Number Enigma

There is an old curious enigma concerning the magical properties of the number 9 which no-one appears to be able to explain (as far as I know), at least not until now.

The enigma is this;

Write down any telephone number. Jumble up the order of the digits. Subtract the smaller number from the larger number. Add together the digits in the answer, and the result will always equal  9.

 

The  answer to any enigma can be found more easily by studying the exception, rather than the rule. In order to solve the enigma we must first look for an exception (if any exists).

 

At first, this example appears to be an exception, the result amounts to 0 and not 9. However, the example is not a true exception to the .sum to  9� algorithm because  it breaks both of the rules of the game � it can be argued that the digits have not been jumbled up and, moreover, that the smaller number has not been subtracted from the larger [because both numbers are of the same hierarchical value, neither is smaller or larger than the other].

This exercise informs us that for the game to prove true, in all cases, the rules of the game .must be strictly adhered to�. Which, in turn, means that the rules are essential factors in the determining the .sum to  9� outcome. This may seem obvious, but it need not be.

GAME RULE 1: The rules of the game must be adhered to.


The Mechanics of the Base-10 numbering system

The number 9 is one of  10 numbers we use in our base-10 numbering system. Our base-10 system orders the accumulation of numbers into columns from left to right, where the column furthest to the right represents the digits that lie from 0-9. Greater accumulations of numbers call for the introduction of another, new, column to the left of the 0-9 column. When this happens, for example when the digits rise in the units column to the limit of 9, a new .tens� column needs to be created to the left. The tens column now indicates 1 and the units column 0 resulting in a  the .decimal� number of 10. Again, when the number of tens exceed 9 another new column needs to be introduced to cope with the larger number and the new column is called the .hundreds� column [for example 100 = 1 in the hundreds column, none in the tens column and none in the digits column]. The opposite applies to reducing numbers that arise due to subtraction. In this case columns reduce in number from the left.

Decimal (Base-10) Rule 1: Greater accumulations of numbers call for the introduction of another, new, column to the left of the 0-9 column and reducing numbers result in the loss of columns from the left. Hence; because there are only nine numbers (and a zero) no number greater than 9 can subsist without the introduction of a new column to the left to accommodate the larger number and no number less than 100 can subsist without the removal of one column from the left.

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The Unique properties of the number 9

Multiples of the number 9  will always compound to 9;

This results because of the use of base 10 as a numbering system;

10-1 = 9   and, considering twice the number:
20-2 = 18, which must equal 9 + 9

In each case, any quantity of the number 9 must always compound to 9

more examples;

30-3 = 27, which must equal 9 + 9 + 9 = 27, 2+7 = 9

Consider an example of a .telephone number 10000;

Decimal (Base-ten) Rule 2: Any quantity of the number 9 must always compound to 9.
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We have seen how the rule works for the number 1000. In fact, it works in the same way for any number greater than 10, using the decimal system. [Numbers less than 10 cannot be considered because an individual number cannot be jumbled prior to subtraction. It is also worth noting that numbers must be jumbled to enable one number to finish up larger or smaller than the other, which would, in turn, enable conformance with the rule of subtracting the smaller from the larger. The jumble-rule is hence, strictly speaking, not a rule in itself but will be considered as such for the purposes of explanation and analysis].
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The effect of .subtracting the smaller number from the resulting larger number�:

Using a simple example of  a .telephone number� of 10;

 

Telephone number 11 - digits cannot be jumbled, hence the number 11 cannot conform with the rules of the game [as was the case with the earlier ostensible exception of 77777].

 

Here, it becomes apparent  that by subtracting the smaller number from the larger number that the base 10 system produces a predictable outcome; this is because the larger number [which in this example (21)] contains two 10s and one 1. The smaller number contains the corollary, one 10 and two 1s. Each time the telephone number increases by 1 its corollary increases by a factor of 10 units.

GAME RULE 2: Digits of the original number must be jumbled.


The effect of the base-10 system when subtracting a smaller number from a larger number

 

When the units column increases by 1 its corollary increases by 10. Because of the rule requiring the smaller number to be  subtracted from the larger the net increase in numbers must always be 10-1 = 9.

Decimal (Base-10)  Rule 4: Each time the telephone number increases by 1 its corollary increases by a factor of 10. Hence, subtraction of the smaller number from the larger number results in a product with a difference of 10-1 = 9.

 

 

 

 

 

 

 

And so on.

But why does this work for large .telephone-size� numbers?  Taking again the example of 10000 used earlier;

Increase by 1

Subtracting smaller from larger

Increase by 1 again;

Subtracting smaller form larger;

Bringing the rules of the game together;

GAME RULE 1: The rules of the game must be adhered to.

GAME RULE 2: Digits of the original number must be jumbled.

GAME RULE 3: Subtracting the smaller number from the larger number within the base-10 system produces a predictable outcome.

together with the mechanics of the Base-10 system;

Decimal (Base-10) Rule 1: Greater accumulations of numbers call for the introduction of another, new, column to the left of the 0-9 column and reducing numbers result in the loss of columns from the left. Hence; because there are only nine numbers (and a zero) no number greater than 9 can subsist without the introduction of a new column to the left to accommodate the larger number and no number less than 100 can subsist without the removal of one column from the left. 

Decimal Base-ten) Rule 2: Any quantity of the number 9 must always compound to 9.

Decimal (Base-10 rule) 3: Subtracting the smaller number from the larger number within the base-10 system produces a predictable outcome.

Decimal (Base-10) Rule 4: Each time the telephone number increases by 1 its corollary increases by a factor of 10. Hence, subtraction of the smaller number from the larger number results in a product with a difference of 10-1 = 9.

Conclusion:

It can be seen that increasing any integer (larger than 10) by 1, jumbling the digits and subtracting the smaller number from the larger number must result in a net difference of 9 between the two sets of numbers. Hence,  because each telephone number is removed by at least 1 unit from its neighbour the outcome of the exercise must apply equally to all telephone numbers. Hence the outcome of the algorithm must always compound to 9. The result is determined by the rules of the game and the mechanics of the base-10 system.

Best wishes = Maurice

 

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