Transfinite Chomp

Summary

ABSTRACT. Chomp is a Nim-like combinatorial game played in or some finite subset. This paper generalizes Chomp to transfinite ordinal space. Transfinite Chomp exhibits regularities and closure properties not present in the smaller game. A fundamental property of transfinite Chomp is the existence of certain initial winning positions, including rectangular positions, and. Many open questions remain for both transfinite and finite Chomp.

Introduction and Notation

In the game of Chomp, cookies are laid out at the lattice points where denotes the natural numbers and play is in dimensions. The cookie at the origin is poisonous. Two players alternate biting into the configuration, each bite eating the cookies in an infinite box from some lattice point outward in all directions, until one player loses by eating the poison cookie. The game can start from a position with finitely many bites already taken from rather than from all of. Chomp was invented by David Gale in [Ga74] and christened by Martin Gardner. When Chomp begins from a finite rectangle it is isomophic to an earlier game, Divisors, due to Schuh [Sch52]. See also [BCG82b], pp.598-606.

This paper considers Chomp on where denotes the ordinals, a subject the first author began studying in the early 1990s. This transfinite version of Chomp has been mentioned in Mathematical Intelligencer columns [Ga93; Ga96]; these are anthologized in [Ga98].

Identifying each ordinal a with the set, the sets

are the boxes at the origin and the Chomp bites in one and two dimensions. Similarly in d dimensions, the Chomp boxes and bites are the corresponding for. Every Chomp position X is a finite union of boxes, and conversely. As we will see, every Chomp game must terminate after finitely many bites.