Kaprekar's constant, 6174

Question

Take any four-digit number that has at least two different digits (leading zeros allowed, so 0031 counts). Arrange its digits in descending order to make the biggest possible number, and in ascending order to make the smallest. Subtract the smaller from the larger. Repeat with the result.

For example, from 3524: 5432 minus 2345 = 3087; then 8730 minus 0378 = 8352; then 8532 minus 2358 = 6174; then 7641 minus 1467 = 6174.

What number does every starting number reach, and what happens once you get there? How many steps can it take at most?

Accepted Answer

Every such number reaches 6,174 within at most 7 steps, and 6,174 then maps to itself forever. The fixed point. 6174 has digits 7, 6, 4, 1. Descending gives 7641 and ascending gives 1467; 7641 minus 1467 = 6174. So once you reach it, you stay. Why everything arrives there. The process only depends on the multiset of digits, and after the first step every result is a multiple of 9 (a number minus a rearrangement of its digits is always divisible by 9, because both have the same digit sum). That collapses the possibilities dramatically. One can check by hand or by computer that among all four-d…

Kaprekar's constant, 6174

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