# Dictionary Definition

chomp n : the act of gripping or chewing off with
the teeth and jaws [syn: bite] v : chew noisily; "The boy
chomped his sandwich" [syn: champ]

# User Contributed Dictionary

## English

### Etymology

U.S. regional variation of “champ” (verb)### Verb

# Extensive Definition

Chomp is a 2-player game played on a rectangular
"chocolate bar" made up of smaller square
blocks (rectangular cells). The players take it in turns to choose
one block and "eat it" (remove from the board), together with those
that are below it and to its right. The top left block is
"poisoned" and the player who eats this loses.

## Example game

Below shows the sequence of moves in a typical game starting with a 3 × 5 bar:Player A must eat the last block and so
loses.

It turns out that for any rectangular starting
position bigger than 1 × 1 the 1st player can win. This can be
shown using a strategy-stealing
argument: assume that the 2nd player has a winning strategy
against any initial 1st player move. Suppose then, that the 1st
player takes only the bottom right hand square. By our assumption,
the 2nd player has a response to this which will force victory. But
if such a winning response exists, the 1st player could have played
it as his first move and thus forced victory. The 2nd player
therefore cannot have a winning strategy.

Computers can easily calculate winning moves for
this game on two-dimensional boards of reasonable size.

## Generalisations of Chomp

3-dimensional Chomp has an initial chocolate bar of a cuboid of blocks indexed as (i,j,k). A move is to take a block together with any block all of whose indices are greater or equal to the corresponding index of the chosen block. In the same way Chomp can be generalised to any number of dimensions.Chomp is sometimes described numerically. An
initial natural
number is given, and players alternate choosing positive
proper
divisors of the initial number, but may not choose 1 or a
multiple of a previously
chosen divisor. This game models n-dimensional Chomp, where the
initial natural number has n prime
factors and the dimensions of the Chomp board
are given by the exponents of the primes
in its
prime factorization.

Ordinal Chomp is played on an infinite board with
some of its dimensions ordinal
numbers: for example a 2 × (ω + 4) bar. A move is to pick any
block and remove all blocks with both indices greater than or equal
the corresponding indices of the chosen block. The case of ω × ω ×
ω Chomp is a notable open problem; a reward has been offered for
finding a winning first move.

More generally, Chomp can be played on any
partially
ordered set with a least
element. A move is to remove any element along with all larger
elements. A player loses if they take the least element.

All varieties of Chomp can also be played without
resorting to poison by using the misère play
convention: The player who eats the final chocolate block is not
poisoned, but simply loses by virtue of being the last player. This
is identical to the ordinary rule when playing Chomp on its own,
but differs when playing the disjunctive
sum of Chomp games, where only the last final chocolate block
loses.

## See also

## External links

chomp in German: Chomp

chomp in Italian: Chomp

chomp in Thai: ชอมป์