For a change of pace, I thought I’d share three interesting mathematical nuggets I’ve run across recently:

Kaprekar’s constant

The first nugget is the magic number 6174, also known as Kaprekar’s constant. Here’s what’s special about it:

1. Take any four-digit number in which at least one digit differs from the rest. For instance, 9998 is OK but 9999 is not. Leading zeros are fine, so 0072 is permitted.

2. Rearrange the digits in descending order. For example, 8953 becomes 9853.

3. Now rearrange the digits in ascending order. 8953 becomes 3589.

4. Subtract the number you got in step 3 from the number you got in step 2. In this case, you get 9853 – 3589 = 6264.

5. Take your new number and perform steps 2-4 on it. You get 6642 – 2466 = 4176.

6. Take that number and repeat steps 2-4.

7. If you keep repeating steps 2-4, you always end up at the number 6174 within at most seven total iterations.

8. Once you arrive at 6174, you stay there forever, because 7641 – 1467 = 6174.

In our example, the entire process looks like this:

Starting number is 8953
9853 – 3589 = 6264
6642 – 2466 = 4176
7641 – 1467 = 6174
6174 is the magic number, so we’re done.

Note that since we’re rearranging digits, this result depends on the fact that we’re using decimal arithmetic — base 10. In other bases, there may or may not be a single magic number.

As you’d expect, Kaprekar’s constant was discovered by a guy named Kaprekar — a schoolteacher from Maharashtra. Kaprekar was a weird dude — he checked all 9,990 possibilities by hand. This sort of thing was his hobby, and he would spend hours doing calculations for fun. He said

A drunkard wants to go on drinking wine to remain in that pleasurable state. The same is the case with me in so far as numbers are concerned.

 

Harmonic series minus 9s

The harmonic series is

That series diverges — that is, even though the terms get smaller, the sum is infinite. However, you can turn it into a convergent series by eliminating all terms that contain the digit ‘9’. So 1/9 is excluded, and so are 1/19, 1/29, etc.

At first glance, it feels counterintuitive that this would convert a divergent series into a convergent one. But it makes more sense when you consider the fact that as the denominator increases, the likelihood of finding the digit 9 somewhere in it increases. Only 1 out of 9 single-digit numbers qualifies (9 itself), but 18 out of 100 two-digit numbers qualify (19,29,…89, 90-99), 271 out of 1000 three-digit numbers qualify, and so on.

Without the 9s, the sum is an irrational number approximately equal to 22.92. It has no known relation to any of the fundamental mathematical constants, such as π or e.

The lonely runner problem

Imagine two runners running around a circular track of length one mile. Assume they are running at different speeds and that they continue running forever, as punishment for being MAGA Republicans. Regardless of their speeds, it is guaranteed that at some point they will be exactly opposite each other. That is, the distance between them will be half the length of the track: 1/2 mile.

A “lonely” runner is defined as one who is at least 1/n miles from every other runner, where n is the number of runners. The runners above become lonely when they are opposite each other — 1/2 mile apart — because n = 2, and 1/n is therefore 1/2.

In the case where n = 3, a runner becomes lonely when they are at least 1/3 mile from each of the other two runners. In the case where n = 4, a runner becomes lonely when they are at least 1/4 mile from the other three runners, and so on.

It has been proven that for values of n from 2 through 10, each runner is guaranteed to become lonely at some point. No one knows whether it is true when there are 11 or more runners.