spectacular nerdularity

[rone]

There's a puzzle in KoTOR which looks like this:

What are the next two steps of this progression? 4, 14, 11-14, 31-14, 13-21-14.

It's enough to drive any math nerd insane.  The answers are 31-13-12-14 and 23-41-12-14.  The trick is to read the progression out loud: 4 is "one four", so you write down 14; then 14 is "one one, one four", 11-14; you read 11-14 as "three ones, one four", 31-14.  So, i got to thinking, "How far can you take this?"

1 11 21 12-11 31-12 13-21-12 31-13-22 23-21-22 42-13-11 14-12-31-13 41-14-12-23 24-31-22-13 32-14-23-21 23-32-21-14 32-23-21-14 (the last two repeat)

2 12 11-12 31-12 [etc.]

3 13 11-13 31-13 23-21 22-13-11 22-31-13 22-23-21 42-13-11 [etc.]

4 14 11-14 31-14 13-21-14 31-13-12-14 23-41-12-14 22-13-24-31 32-21-23-14 23-32-21-14 32-23-21-14 (the last two repeat)

For any other digit n, it's:

n
1n
11-1n
31-1n
13-21-1n
31-13-12-1n
23-41-12-1n
22-13-14-31-1n
22-41-23-14-1n
32-24-31-13-1n
33-22-14-31-1n
33-22-31-14-1n
(repeats)

So... what conclusions can we extract?  The words of Gregg Easterbrook's alter ego, Tuesday Morning Quarterback, come to mind:

This is the kind of hidden indicator that is essential to an insider's understanding of the game. Unfortunately, Tuesday Morning Quarterback has no idea what it means.

Comments

[doctroid.livejournal.com]

Ah, I see why I was initially confused by your series. I've seen this puzzle before, but your interpretation is slightly different: you go

  4            ("one four")
  14           ("one one, one four")
  11-14        ("three ones, one four")
  31-14        ("one three, two ones, one four")
  13-21-14     ("three ones, one three, one two, one four")
  31-13-12-14
  ...

whereas the version I've seen diverges from this in the last step shown:

  4            ("one four")
  14           ("one one, one four")
  11-14        ("three ones, one four")
  31-14        ("one three, two ones, one four")
  13-21-14     ("one one, one three, one two, two ones, one four")
  11-13-12-21-14
  ...

Needless to say the series evolve quite differently from there.

[damienw]

right: R's string is independent of position, whereas yours (and the more usual one, I think) is context dependent. There's a great deal of work on these sorts of numbers, although I think most of it these days is in grammars and theoretical computer science... I certainly don't know much about them.

[nothings.livejournal.com]

The best part about this is that it diverges at exactly the point where the puzzle is phrased. So the puzzle in the game is broken--there are two reasonable answers! (I figured it out pretty fast, or thought I had, except that, hey, my answers were 'your' answers instead of rone's answers. Also dumplechan's comment below is using this 'wrong' version.)