## Misere quotients for impartial games

Elwyn Berlekamp suggested that the title of my recent paper with Aaron Siegel,* The Phi-values of various games*, be changed:

I urge you to change the title. I think you have a good chance to make an impact well beyond the small circle of experts such as those who come to conferences like the one in Banff. There are many more mathematicians and algebraists out there, as well as a big computer Go community. Terms like “Phi-function” are unlikely to become buzzwords that stick, as there are many Phi functions defined by many different authors. “Misere Quotient” might well be a unique pair of words. It could quite plausibly become buzzword that would help stir up more interest in combinatorial games in general. So I think it important that your title include that pair of words. I’d propose a short title such as “Misere Quotients for Impartial Games”.

It’s hard to argue with that, and it’s still a preprint. So Misere Quotients for Impartial Games it is!

## The Phi-values of various games

Just in time for Richard K. Guy’s 90th birthday a few days ago (30 September 2006), Aaron Siegel and I finished our paper The Phi-values of various games.

It just appeared on the arXiv. Check it out!

## MisereSolver

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.

## Allemang’s thesis

**Machine Computation with Finite Games**

In 1984, Dean Allemang wrote an M.Sc. thesis at Cambridge University that is today [January 2003] still an excellent resource for results on misère octal games. The title of his thesis is *Machine Computation with Finite Games*.

In many cases the contents of Allemang’s thesis go beyond what is contained in Chapter 13 of the 1985 (third printing) of *Winning Ways* (“the last and most complicated theory in this book,” according to the authors, Berlekamp, Conway, and Guy).

The original electronic version of Allemang’s thesis has been lost. After cajoling Dean into sending me a photocopy, I scanned it in and converted it to PDF.

I’ve split the document into two parts. Here’s a rough outline of the contents.

Part I [PDF, 48 pages, 2.56 MB]

Cover

Declaration by Author

Abstract

Misère Play of Impartial games

Normal play

Misère play

Conway’s Cancellation

The Genus Sequence Algorithm

Grundy’s Game

Other Games

Tins

Guiles

The octal game .17

Errors inWinning Ways

A Bigger Genus Sequence

The Generalized Genus Statement

Periodicity in Genus Sequences

An Algorithm for Generalized Genera

Errors inWinning Ways

The Game of Knots Ties Up all Genus Sequences

Which sequences arise as genus sequences?

Part II [PDF, 33 pages, 15.2 MB]

Complete Analysis for Misère Games

A Catalogue of Octal Games

Appendix III: “An implementation of many of the computations necessary for the analysis of normal partizan games, as treated in Conway,On Numbers and Games…”.