Chomp is a 2-player game played on a rectangular chocolate bar made up of smaller square blocks. The players take it in turns to choose one block and eat it, together with those that are below it and to its right. The top left block is poisoned and the player who eats this loses. A square as a geometric shape is described and illustrated at square (geometry). ...

Example game

Below shows the sequence of how a typical game could look starting with a 3x5 block

Player A must eat the last block and so loses.

Who wins?

Chomp belongs to the category of 2-player perfect information games and so any position can be said to be a 1st player win or a 2nd player win. Perfect information is a term used in economics and game theory to describe a state of complete knowledge about the actions of other players that is instantaneously updated as new information arises. ...

It turns out that for any rectangular starting position bigger than 1 x 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. In combinatorial game theory, the Strategy-stealing argument is a general argument that shows, for many games, that the second player cannot have a winning strategy. ...

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 which all of whose indices are greater or equal to the chosen block. In the same way Chomp can be generalised to any number of dimensions. Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ... In anatomy, the cuboid bone is a bone in the foot. ...

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. Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), and they can be used for ordering (this is... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... In mathematics, a divisor of an integer n, also called a factor of n, is an integer which evenly divides n without leaving a remainder. ... Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ... In number theory, the prime factors of a positive integer are the prime numbers that divide into that integer exactly, without leaving a remainder. ... Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ... In mathematics, exponentiation is a process generalized from repeated (or iterated) multiplication, in much the same way that multiplication is a process generalized from repeated addition. ... In mathematics, and in particular number theory, the fundamental theorem of arithmetic or unique factorization theorem is the statement that every positive integer greater than 1 is either a prime number or can be written as a product of prime numbers. ...

Ordinal Chomp is played on an infinite board with some ordinal as its length. For example a 2 by ω + 4. A move is to pick any block and remove all blocks with both indices greater than or equal to it. The case of ω by ω by ω Chomp is a notable open problem; a reward has been offered for finding a winning first move. Ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ...

More generally, Chomp can be played on any partially ordered set with a least element. A move is to remove any element along with any larger elements. A player loses if they take the least element. In mathematics, especially order theory, a partially ordered set (or poset for short) is a set equipped with a partial order relation. ... In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually. ...

All varieties of Chomp can also be played without resorting to poison by using the misère play convention: The player who eats the final cookie 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 cookie loses. A misère version of a game is a game that is played according to its conventional rules, except that it is played to lose; that is, the winner is the one who loses according to the normal game rules. ... The disjunctive sum of two games is a game in which the two games are played in parallel, with each player being allowed to move in just one of the games per turn. ...

See also

• Nim

Nim is a two-player mathematical game of strategy in which players take turns removing objects from heaps, one or more objects at a time but only from a single heap. ...

External links

• More information about the game
• A freeware version for windows
• Play chomp online