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.