The following are two of the mathematical BLACK HOLES...

The Sisyphus String: 123

Suppose we start with any natural number, regarded as a string, such

as 9288759.

Count the number of even digits, the number of odd digits, and the

total number of digits. These are 3 (three evens), 4 (four odds), and

7 (total of seven digits). So, use these digits to form the next

string or number, 347.

Now repeat with 347, counting evens, odds, total number, to get 1, 2,

3, so write down 123.

If we repeat with 123, we get 123 again.

The number 123 with respect to this process and universe of numbers is

a mathematical black hole.

All numbers in this universe are drawn to 123 by this process, never to escape.

Will every number really be sent to 123? Try

122333444455555666666777777788888888999999999.

The numbers of evens, odds, and total are 20, 25, and 45,

respectively. So, our next iterate is 202545, the number obtained from

20, 25, 45. Iterating for 202545 we find 4, 2, and 6 for evens, odds,

total, so we have 426 now. One more iteration using 426 produces 303,

and a final iteration from 303 produces 123.

Kaprekar's constant (number): 6174

Process in this can be described as follows.

Take any four digit number with distinct digit, form a number by

arranging these digits in ascending order and subtract this number

from a number formed by arranging same digits in descending order. Now

repeat the procedure on the subtraction. You will finally reach to

number 6174.

eg. Take no. 1234

4321 - 1234 = 3087

8730 - 0378 = 8352

8532 - 2358 = 6174

7641 - 1467 = 6174

Regards,

Subhash Kasundra

TATA Consultancy Services Ltd.

Ahmedabad.

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