Let’s play a short game:

1. Choose a four-digit number. (The only condition is that it has at least two different digits.)

2. Arrange the digits of the four-digit number in descending then ascending order.

3. Subtract the smaller number from the bigger one.

4. Repeat.

Do you observe anything strange? If not convinced, try out with a different number. You will arrive at the same result.

You make try it out with any combinations as per the rules mentioned above. Eventually, you’ll end up at 6,174. This number 6,174 is known as the Kaprekar’s constant.

Video Courtesy – ” Numberphile ”

**History**

In 1949 the mathematician D. R. Kaprekar from Devlali, India, devised a process now known as *Kaprekar’s operation*. First choose a four digit number where the digits are not all the same (that is not 1111, 2222,…). Then rearrange the digits to get the largest and smallest numbers these digits can make. Finally, subtract the smallest number from the largest to get a new number, and carry on repeating the operation for each new number.

It is a simple operation, but Kaprekar discovered it led to the above surprising result.

** Working Rule**

The digits of any four digit number can be arranged into a maximum number by putting the digits in descending order, and a minimum number by putting them in ascending order. So for four digits *a,b,c,d* where

*9 ≥ a ≥ b ≥ c ≥ d ≥ 0*

*a, b, c, d*are not all the same digit, the maximum number is

*abcd*and the minimum is

*dcba*.

We can calculate the result of Kaprekar’s operation using the standard method of subtraction applied to each column of this problem:

abcd

– dcba

—————–

ABCD

which gives the relations:

D = 10 + d – a (as a > d) |

C = 10 + c – 1 – b = 9 + c – b (as b > c – 1) |

B = b – 1 – c (as b > c) |

A = a – d |

for those numbers where *a>b>c>d*.

A number will be repeated under Kaprekar’s operation if the resulting number *ABCD* can be written using the initial four digits *a,b,c* and *d*. So we can find the kernels of Kaprekar’s operation by considering all the possible combinations of {*a, b, c, d*} and checking if they satisfy the relations above. Each of the 4! = 24 combinations gives a system of four simultaneous equations with four unknowns, so we should be able to solve this system for *a, b, c* and *d*.

It turns out that only one of these combinations has integer solutions that satisfy *9 ≥ a ≥ b ≥ c ≥ d ≥ 0*. That combination is *ABCD = bdac*, and the solution to the simultaneous equations is *a*=7, *b*=6, *c*=4 and *d*=1. That is *ABCD*= 6174. There are no valid solutions to the simultaneous equations resulting from some of the digits in *{a,b,c,d}* being equal. Therefore the number 6174 is the only number unchanged by Kaprekar’s operation — our mysterious number is unique.

**Interesting Corollary**

For three digit numbers the same phenomenon occurs. For example applying Kaprekar’s operation to the three digit number 753 gives the following:

753 – 357 = 396

963 – 369 = 594

954 – 459 = 495

954 – 459 = 495

The number 495 is the unique kernel for the operation on three-digit numbers, and all three-digit numbers reach 495 using the operation. Why don’t you check it yourself?

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