6174 is known as Kaprekar’s Constant. Why is this number important? Perform the following process (called Kaprekar’s Routine):

- Take any two digit number whose digits are not all identical.
- Arrange the digits in descending and then ascending order to get two four digit numbers.
- Subtract the smaller number from the bigger number.
- Go to step 2 and repeat.

This process will always reach its fixed point **6174** in at most 7 iterations. 6174 is a fixed point as once it has been reached, the process will continue yielding 7641 – 1467 = 6174.

### Example: 3141

4311-1134=3177

7731-1377=6354

6543-3456=3087

8730-0378=8352

8532-2358=6174

7641-1467=6174

### The Maths Behind it

Each number in the sequence uniquely determines the next number. As there are only finitely many possibilities, eventually the sequence must return to a number it has already hit. This leads to a cycle.

So any starting number will give a sequence that eventually cycles.

There can be many cycles, but for 4 digit numbers in base 10, there happens to be 1 non – trivial cycle, which involves the number 6174.

To read more, click here.

### Numberphile Video

Click here for my previous post about Kaprekar. M x