Take any four digit number (whose digits are not all identical), and do the following:

- Rearrange the string of digits to form the largest and smallest 4-digit numbers possible.
- Take these two numbers and subtract the smaller number from the larger.
- Use the number you obtain and repeat the above process.

What happens if you repeat the above process over and over? Let's see…

Suppose we choose the number 3141.

4311-1134=3177.

7731-1377=6354.

6543-3456=3087.

8730-0378=8352.

8532-2358=6174.

7641-1467=6174…

The process eventually hits 6174 and then stays there!

But the more amazing thing is this: *every* four digit number whose digits are not all the same will eventually hit 6174, in at most 7 steps, and then stay there!

**Presentation Suggestions:**

Remember that if you encounter any numbers with fewer than has fewer 4 digits, it must be treated as though it had 4 digits, using leading zeroes. Example: if you start with 3222 and subtract 2333, then the difference is 0999. The next step would then consider the difference 9990-0999=8991, and so on. You might ask students to investigate what happens for strings of other lengths or in other bases.

**The Math Behind the Fact:**

Each number in the sequence uniquely determines the next number in the sequence. Since there are only finitely many possibilities, eventually the sequence must return to a number it hit before, leading to a cycle. So any starting number will give a sequence that eventually cycles. There can be many cycles; however, for length 4 strings in base 10, there happens to be 1 non-trivial cycle, and it has length 1 (involving the number 6174).

**How to Cite this Page:**

Su, Francis E., et al. “Kaprekar's Constant.” *Math Fun Facts*. <https://www.math.hmc.edu/funfacts>.

**Fun Fact suggested by: **

Byron Walden