Some commutative monoids for specific games are now online for you to browse, if you like.
There’s more to come!
Aaron Siegel has found many more wild misere game solutions amongst the four-digit octals, including this amazing one, for 0.4107.
Misere game solution, or extraterrestrial transmission?
Aaron Siegel writes:
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0.4107 has period 24, preperiod 66, and quotient order 506 – the largest yet discovered! Its quotient has a minimal set of 34 generators (!!) It’s quite neat to look at; there are many irregular values among the smaller heaps, until finally a pattern suddenly emerges. This behavior makes me wonder how many other super-complicated-looking games nonetheless settle down fairly quickly. I can’t resist including the pretending function for 0.4107 🙂
Phi = 1 1 a a b b ab c c d e f g h b i ab2 j k l m n o p q r abo anq b3 s t abm cq2 u cjk v w x b3 y agt z b2i A B b3 C D b4c bco abF E b3 F ab3c grx G abF abF b3 ab3c H b3 b4c ab4 cfH b4c ab3c ab3c b3 b3 ab4 b4c b4c ab4 ab3c b3 b3 ab3c b4c b4c ab4 ab4 b3 ab3c ab3c b3 b4c ab4 ab4 b4c ab3c ab3c b3 b3 ab4 b4c b4c ab4 ab3c b3 b3 ab3c b4c b4c ab4 ab4 b3 ab3c ab3c b3 b4c ab4 ab4 b4c ab3c ab3c b3 b3 ab4 b4c b4c ab4 ab3c b3 b3 ab3c b4c b4c ab4 ab4 b3 ab3c ab3c b3 b4c ab4 ab4 b4c ab3c ab3c b3 b3 ab4 b4c b4c ab4 ab3c b3 b3 ab3c b4c b4c ab4 ab4 b3 ab3c ab3c b3 b4c ab4 ab4 b4c ab3c ab3c b3 b3 ab4 b4c b4c ab4 ab3c b3 b3 ab3c b4c b4c ab4 ab4 b3 ab3c ab3c b3 b4c
Just look at that cluster of anomalies – “j k l m n o p q r” – that all smooth out in the end . . .
Solutions to wild quaternary games PDF (Aaron Siegel).Aaron has also found a solution for the wild octal game 0.144.
More tidbits from Aaron:
0.01222 has period 8 and quotient order 370
0.10322 has period 8 and quotient order 214
Aaron Siegel has found a complete analysis of the misere play of Guiles (Guy and Smith octal code 0.15) by using his recently developed standalone java program, MisereSolver.
The misere quotient semigroup of Guiles has 42 elements, and its pretending function has period 10.
There will be a paper coming out in the next year or so that gives more information. If you want to wade through the details now, before it’s cleaned up for publication, here’s the solution file that gives its misere indistinguishability quotient semigroups to heap 160. That’s more than long enough to guarantee that the periodicity continues indefinitely. And here’s an even bigger log file documenting the same computation, also to heap 160.
Aaron Siegel has already whipped together an amazing Java program for everything in my misere CGT paper, while also incorporating his own insights into algorithms for misere quotient semigroup computation. Given an octal game code as input, his program directly computes a presentation of its misere quotient semigroup to heap size n=1, 2, 3, in turn, together with the associated pretending functions and outcome partitions at each heap size n. He’s solved .15 (Guiles), .115, .114, and a bunch of wild quaternary games using this software.
Eventually his software will make into cgsuite, I hope.
Aaron Siegel has claimed 17 of the 21 quaternary bounties. Still open are .3102, .3122, .3123, and .3312. That’s $425 in bounties for Aaron, but there’s still $100 on the table…
Quaternary Bounties (PDF, 2 pages)
I’m hoping to contribute some software for misere games for incorporation into cgsuite.
A short note on the quotient semigroup viewpoint for misere games. I wrote this note two days after discovering the quotient semigroup construction.