in Mathematics

The mysterious and beautiful 6174

My favorite will always be 1.618 (The Golden Ratio) but 6174 fascinated me recently mainly because of the mystery it carries.

It was in 1949 when an Indian┬ámathematician named D. R. Kaprekar, devised a process now known as Kaprekar’s operation. This was published in “Scripta Mathematica” which was a quarterly journal published by Yeshiva University. This was the only magazine which was believed to be edited by masters for laymen at that time.

So whats the big deal?┬áPick any four digit number where all the numbers are not same for example 2500. Now rearrage the digits to get the largest and the smallest number that it can make. Subtract the smallest number from the largest to get a new number, repeat this operation for every new number. All this seems simple isn’t it? But how is it a beautiful pattern? Karpekar discovered that this process actually led to a surprising result.

Lets take 2500 for instance and see how it turns out to be beautiful. The smallest number that you can make up with 2500 is 25 and the largest being 5200.

5200 – 0025 = 5175
7551 – 1557 = 5994 (Made from 5175 the result of previous subtraction)
9954 – 4599 = 5355
5553 – 3555 = 1998
9981 – 1899 = 8082
8820 – 0288 = 8532
8532 – 2358 = 6174
7641 – 1467 = 6174

When we reach 6174 the operation repeats itself, returning 6174 every time. Karpekar called it the kernel of this operation.

Lets try another number,

9990 – 999 = 8991
9981 – 1899 = 8082
8820 – 288 = 8532
8532 – 2358 = 6174 ( Reached 6174 again! )

Why is it special?
Every 4 digit number combination will end up in 6174.

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