I got 99 solutions

A few days ago on Reddit, I saw someone post this observation: "The number 14233221 describes itself; it has one four, two threes, three twos, and two ones." I thought that was pretty interesting, so I thought I'd take a closer look. The numbers are very similar to the look-and-say sequence as studied to death by John Conway of Game of Life fame.

In the traditional look-and-say sequence you say 1, 11, 21, 1211, 111221, or "one", "one one", "one two, one one", and so on. However, we want to count the digits instead of reading them in order, so: 11, 21, 1112, 3112, [...], 21322314. This last number is actually the same as our original 14233221 with the digits in a different order. These have been described as descriptive numbers and counting sequences. What we want are the fixed points of this descriptive/counting function.

Another closely related idea is the autobiographical or self-descriptive number. These are numbers such as 1210 where the first digit represents the number of 0s, the next the number of 1s and so on. The largest of these is 6210001000. In fact, there are only 7 of them assuming you only allow digits up to 9. However, there are many more of the kinds of numbers we're looking for.

To find out just how many, I put together a program to search through them. Initially I wanted to be clever and search in some incremental way through the space of numbers by just picking some seed numbers and generating new numbers until I found the fixed points. Unfortunately, I couldn't figure out how to be sure that I'd get all the numbers. Maybe someone with more substantial mathematical chops than me would be able to figure it out, but I had to settle for a brute force approach.

However, I found an interesting shortcut: while you can have a solution that starts with 10 (like 10213223), you can't have one that starts with 20, or indeed any number of 0s other than 1. This eliminates 80% of the candidate solutions fairly quickly. I maybe could have found other clever things, but by that point my program was fast enough anyway, so I just ran it. It turns out the smallest of these numbers is 22, and there is only one solution that uses every digit. I'll put the full list in a box below, but it might be a fun exercise to find the largest one yourself.

One curious fact: there are, in fact, 99 solutions. Since there does not appear to already be a satisfying name for these, how do we feel about Beyoncé numbers?

22 10213223 10311233 10313314 10313315 10313316 10313317 10313318 10313319 21322314 21322315 21322316 21322317 21322318 21322319 31123314 31123315 31123316 31123317 31123318 31123319 31331415 31331416 31331417 31331418 31331419 31331516 31331517 31331518 31331519 31331617 31331618 31331619 31331718 31331719 31331819 1031223314 1031223315 1031223316 1031223317 1031223318 1031223319 3122331415 3122331416 3122331417 3122331418 3122331419 3122331516 3122331517 3122331518 3122331519 3122331617 3122331618 3122331619 3122331718 3122331719 3122331819 10413223241516 10413223241517 10413223241518 10413223241519 10413223241617 10413223241618 10413223241619 10413223241718 10413223241719 10413223241819 41322324151617 41322324151618 41322324151619 41322324151718 41322324151719 41322324151819 41322324161718 41322324161719 41322324161819 41322324171819 1051322314251617 1051322314251618 1051322314251619 1051322314251718 1051322314251719 1051322314251819 1051322325161718 1051322325161719 1051322325161819 1051322325171819 5132231425161718 5132231425161719 5132231425161819 5132231425171819 5132232516171819 106132231415261718 106132231415261719 106132231415261819 106132231426171819 106132231526171819 613223141526171819 10713223141516271819