Sunday, April 5, 2009


Take a four digit number. Put the digits in order from smallest to biggest to make number A. Then put them in order from biggest to smallest to make number B. Calculate B minus A to get the answer.

Repeat this process with the answer, and keep repeating.

For example start with 7119.
Putting the digits in order from smallest to largest:
Now from largest to smallest:
9711 - 1179 = 8532

Now repeat with 8532:
8532 - 2358 = 6174

Now repeat with 6174:
7641 - 1467 = 6174

We've reached the end since applying this process to 6174 just gives 6174 back again.

If fact, no matter where you start, if you keep going, you will get to 6174. (Unless all the original digits were the same. Then you will get zero on the first calculation.)

How does this work?

It turns out that whatever number you start with, that after doing the process the first time you will be left with one of 54 numbers (or zero). From there it is at most 6 steps away from 6174.

Here is a graph of those 54 numbers. Start with a number and follow the arrow to see what you get after applying this process once. Keep following the arrows down to 6174.

Click on the graph to enlarge it.

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