Project III (MATH3131) 2015-16

On Numbers and Games

Herbert Gangl

Description

Mathematical (2-person) games can be analyzed by attaching to each game position a "number" in an ingenious way. Depending on the games, the corresponding game positions can then be identified with numbers, in the simplest cases rational ones, but it is easy to produce games which inevitably lead to irrational game position values. In this way, one can view every real number as (the value of) certain game positions. Moreover, one quickly stumbles over extensions of the usual number system which produce infinitesimal or even "fuzzy" values.

John H. Conway [1] has invented this interplay between numbers and games, and it has been joyfully developed by him in collaboration with Berlekamp and Guy in [2] which provides a wealth of beautiful examples of very tempting games. He who knows how to quickly evaluate the positions in a given game has the best chances to win it.

The goal of the project would be on the one hand to explore the different kinds of numbers that can be produced in such mathematical games, and on the other to analyze the games at hand as well as perhaps to invent new ones. Maybe some students could even write an app that produces the best possible moves for any position in certain games?

Prerequisites

There are no particular prerequisites required, but knowledge of ordinal numbers will be helpful.

Resources

The following book is the original (and standard) reference

[1] J. H. Conway On numbers and games .
with a short survey in the American Mathematical Monthly All games bright and beautiful

Presumably the best starting point is

[2] E. Berlekamp, J.H. Conway, R. Guy Winning ways for Your Mathematical Plays I+II

Take a glimpse (a few pages online via Google Scholar).

There is a more informal introduction of the numbers involved by Donald Knuth (the author of the TeX program which you will wind up using for text processing) what he calls

D. Knuth Surreal numbers,

and a somewhat more sober version is given by H. Hermes in Chapter 13 of

Numbers (Graduate Texts in Mathematics / Readings in Mathematics) by Friedrich Hirzebruch et al.

Here is the chapter we discussed last time.

Here is the chapter we talked about today.

email: Herbert Gangl

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