Dan Hoey sent me this interesting page he’s put together on the game of Nim in the screenplay of the movie Last Year at Marienbad.
Since I’m going to this math event in Halifax in less than a month, I thought I should spend some time thinking about the topic of misere partisan games, since that’s the only thing mentioned as a definite topic. I worked on it today at the loft, as the kids played Katamari Damacy on the PS/2.
As usual, I can’t be bothered to do any heavy lifting and I don’t have any particularly good ideas except a healthy suspicion that the whatever I first think about the subject will turn out to be wrong, as well as the second, third and fourth thing I think about it. But I’m always willing to do some computer programming, or fiddle around with other calculations.
Here are the results of a Mathematica search I did today for pairwise mutually distinguishable partisan misere games at birthday two. At birthday 0 and 1, there are 1 and 4 mutually distinguishable normal and misere games (ie, 0, 1, -1 and star, ie *1). But at birthday two, there are only 22 normal play games (I think), but at least 123 misere ones, if this program is correct. There are probably more distinguishable types. There are at most 256.
I’m getting a headache trying to figure out the correct canonical form reduction rule in the monoid of all partisan misere games, so I think I’ll forget about that for now. Also, even thinking about order relations is giving me pain.
It’s a start at least.
I just put a paper into the arXiv called Advances in Losing. It’s actually been done for awhile; I just got around to doing it. Now I’m working on a paper with Aaron Siegel called “The Phi-values of various games,” and I needed to have a reference to this other paper, so I put it into the arXiv. “Advances in Losing” is eventually going to be in a Cambridge Univ Press book being edited by Richard Nowakowski and Michael Albert called Games of No Chance 3.
A new version of the game to be played on the logo of the 2006 Gathering for Gardner (G4G7) meeting invitation, fixing errors pointed out by Dan Hoey [thanks, Dan!]
Moments in Last Year at Marienbad when that weird character plays misere Nim against the other hotel guests
Last Year at Marienbad is a 1961 French movie directed by Alan Renais. It’s mostly annoying to watch, but it has the benefit of having the game of Misere Nim played several times in the movie.The starting position is always four heaps of cards (more often, matchsticks). The heaps have sizes 1, 3, 5, and 7. A move is to take some number of cards from one heap only, removing them from play (including the whole heap, if desired). Play ends when the last card is taken, and the player who takes that card loses the game. This starting position is a forced win for the second player to move (ie a P-position) in best play.
Last night, I suffered through the whole movie, keeping track of when Misere Nim is played.
#1) Elapsed time: 15:11 (roughly): Misere nim with cards. Best view of play in the movie. As always in the movie, the strange-looking guy who proposes that the game be played wins. He lets the other guest move first. The actor playing the Guest does a great job of looking disgruntled as he is forced to take the last card (and loses the game).
#2) Elapsed time: 20:56: Misere nim with matchsticks. “What if you play first?” The weird guy obliges, and makes a (losing) first move, but the guest makes an error and the weird guy wins again.
#3) Elapsed time: 37:00: Misere nim with matchsticks, set up for play only.
#4) Elapsed time: 1hr 13:00: Misere nim with matchsticks. This is the best part, with the highly amusing speculation on the part of the guests—”I think you should always take an odd number.” “He’s using the theory of logarithms.” [In translation to English on my DVD, that is “It is a type of logarithmic series.” “How does he always win?” This last scene is the best one for commentary by the guests.
I like this review of the movie at Amazon, by Jack Walter:
I am an avid fan of foreign, avant-garde, bizarre, challenging and/or enigmatic films, but this one is just plain agonizing to watch. The photography and the characters are beautiful, but I had to view this film in two sessions, both of them tormentingly slow. At first I thought it was some kind of variation on Sartre’s “No Exit,” but if it was, I was the one in the waiting room in Hell! This movie is pointless, vapid and pathetically pretentious. I hope God adds ninety-four minutes onto my life as a reward for sitting through Last Year at Marienbad!
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:
* * * *
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.