dtm: (Default)
[personal profile] dtm
I wan't sure when I wrote the previous post, but now I am: there are in fact two distinct Rock-Paper-Scissors 7-degree extensions.

Any 7-degree RPS extension can be turned into a 5-degree RPS extension by declaring two choices unavailable to players, so one way to approach things is to start with (the only) 5-token RPS game and add two tokens to it. (*)

So let's start with tokens 0, 1, 2, 3, 4, with token i winning against tokens (i+1) and (i+2) and losing against (i-1) and (i-2), all arithmetic happening modulo 5. Now, to this batch we add two tokens, call them "a" and "b". WLOG, assume that b beats a. Then, a must beat exactly three of {0,1,2,3,4}. Up to renaming those five tokens, we have two choices: either the three tokens a beats are consecutive {0,1,2} or not {0,1,3}.

So we know there are at most two different possibilities for a 7-degree RPS. If a beats {0,1,2}, then you can arrange the tokens in a circle of {0,1,2,b,3,4,a} and each token will beat the tokens 1, 2, or 3 spots to the right (wrapping around), and lose to the tokens 1, 2, or 3 spots to the left. Note that the three tokens that a beats are not, by themselves, something you can play a good RPS game with since 0 beats both 1 and 2. (Imagine playing RPSSL with just rock, paper, and Spock) In fact, by the symmetry of this game, the set of tokens beaten by any given token would not make a good set of symbols to play RPS with.

However, if a beats {0,1,3}, then the set of symbols a defeats do form a set you could play RPS with, so the two games really are different and aren't just renamed versions of each other.

(*) Oh crud. I didn't justify this statement, and I'm suddenly not sure it's true. It's certainly not true that you can take a degree-p RPS extension, cut off (p-q) nodes at random, and automatically get yourself a degree-q extension for any p and q. So I guess we have "at least two" degree-7 RPS extensions.
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