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In the game of Chomp, cookies are laid out at the lattice
points $N^d$ with a poison cookie at the origin. Two players alternate
biting into the configuration, each bite eating the cookies in an
infinite box from some point outward in all directions, until someone
loses by eating the poison cookie. The game can also start with
finitely many
bites already taken. This talk discusses Chomp extended to $\Omega^d$
(where $\Omega$ denotes the ordinals), a subject investigated by Scott
Huddleston. Transfinite Chomp has satisfyingly complete results. In
two dimensions, for example, a position of two columns of equal height
$h$ loses for the player who reaches it if $h\in Z^+$, but it wins for
the player who reaches it if $h=\omega$, and then it loses again for
the player who reaches it if $h>\omega$. For three columns of height
$h$, the position is a win for the player who reaches it if and only if
$h=\omega^\omega$. In three dimensions, the box of height $h$ over a
2-by-2 base is a win for the player who reaches it if and only if
$h=w^3$.
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