Calculator Games: The Kaprekar Constant

The Kaprekar Constant: 6174

This is one of those problems that doesn't come from Calculator Fun and Games, but would be an interesting offshoot, and can certainly be done with a calculator, pen and pencil.

Take any four digit number.  It must contain at least two different digits (i.e. 9933 is permitted, but 8888 is not).  Sequence the digits in descending order (e.g. 2536 becomes 6532), and ascending order (2536 becomes 2356).  Subtract the smaller number from the larger, then repeat the process of finding the descending and ascending numbers, and subtracting.  To quote Calculator Fun and Games (and countless vague Maths GCSE questions):  What do you notice?

First example:

1874
Rearrange:  8741-1478 = 7263
Rearrange: 7632 - 2367 = 5265
6652 - 2566 = 4086
8640 - 0468 = 8172
8721 - 1278 = 7443
7443 - 3447 = 3996
9963 - 3699 = 6264
6642 - 2466 = 4176
(7641 - 1467 = 6174, which goes round in a loop)

Some wider reading shows that 6174 is the Kaprekar constant, and it is the end result for all four-digit numbers, and that all numbers terminate there after no more than seven steps.  Never one to shirk a challenge (especially if it involves a calculator), I tried out some more numbers:

9954 --> 5553 --> 9981 --> 8870 --> 8532 --> 6174

9732 --> 7533 --> 6174

1110 --> 9900 --> 9810 --> 9621 --> 6174

I repeated this several times, with upwards of 30 different numbers, and in order to summarise my results, I have drawn out a network of key paths to 6174.  I've consistently used a colour scheme - dark green for 6714, light green for numbers which lead directly to 6174; yellow for numbers which need two steps, light blue need three steps, and so on.

Firstly:  pathways for numbers which require all seven steps to reach 6174.  I've found four numbers that require seven steps, and they follow the paths below; three of them go through 9954, and they all go through 8532.  My work on the Kaprekar constant tried to focus on numbers which either need all seven steps, or resolve to 6174 in one step.

And interestingly, five six-step numbers that all follow the same pathway:

And finally, a collection of one-step and two-step numbers which have no parents:

One step:  6642, 8532, 9621, 7533, 7531, 6200, 8754 and 8752.  Bold numbers have no parents - no other numbers (that I have found) go through them to 6174.  7533 has six parents, but none of these parents have any parents of their own.

These findings are all based on my own work, and I fully acknowledge that they could be incorrect due to my incomplete research.


Extension:  a five-digit Kaprekar process... another time!