The ‘magic’ number 495

Today I want to talk about an interesting mathematical curiosity.

First, I want you to think of any three digit number where all the digits are different.

I will work through this example using my number of 729.

What I want you to do is the following routine:

1. Rearrange the digits to make the largest and smallest numbers
2. Subtract the smaller number from the larger one
3. For the resulting number, repeat the procedure, and keep repeating

For example, with my number of 729, the routine says:

1. Make the largest number (972) and the smallest number (279)
2. Subtract the smaller number from the larger (972 – 279 = 693)
3. Repeat the procedure for 693, and keep repeating.

What happens is the following:

In this example, notice we end up with the number 495, and that number just goes to itself.

The remarkable thing is that this happens for every three digit number where the digits are distinct.

In fact, it even happens for numbers like 7 if you add on leading zeros (so 7 is really 007, the numbers you can make are 700 and 007. Subtracting leads to 693, which we showed above ultimately results in 495).

The only numbers that do not work are ones where all the digits are the same like 111, 222, and so on, as they obviously result in the number 0.

The Kaprekar routine

Why does the number 495 result from this process?

This graph proves exactly how all the three digit numbers end up as 495.

(click on picture for larger image)

(image from WikiMedia Commons, CC by 3.0)

The history of this discovery is actually about a more complicated case. In 1949, the Indian mathematician D.R. Kaprekar found the exact same thing happened for four digit numbers, and the result of that was 6174. (I will leave this to you to try out).

Why this happens is explained in details in this excellent article from Plus magazine.

The short explanation is the Kaprekar routine, as the transformation is now known, leads to these numbers as a bit of mathematical curiosity. That is, it just so happens the routine results in a single number for three digits and four digits.

If you move to five digits, for example, the Kaprekar transformation does not result in a single number. Instead, you could end up with any of the 10 numbers: 53955, 59994, 61974, 62964, 63954, 71973, 74943, 75933, 82962, 83952. People have extended the work into other number base systems as well, but there is not always a single fixed point (see this).

The connection with game theory

I didn’t put this article on Game Theory Tuesdays by accident.

The connection I wanted to make is that 495 is an example of a fixed point, a point that stays the same after the numerical operation of the Kaprekar routine.

Fixed points are a useful mathematical entity, and they have a special place in game theory. The way John Nash proved that many games have equilibria is by using a fixed point theorem: he showed that under certain conditions, there must exist a set of strategies such that no player can deviate profitably. The set of strategies is a fixed point that survives the strategic thinking of every player responding to the other players. (more on that here)

Unlike other fixed point examples that depend on calculus, the Kaprekar fixed point is a simple and beautiful one that depends on plain arithmetic, and that is why I like it as a teaching tool.

Ultimately this is just a mathematical curiosity, but I feel it is one that can exemplify fixed points and help students relate to the material.