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In combinatorial game theory, a cold game is a position with no gain from taking the next move, as illustrated by number games. For $G=\{0\mid 1\}=1/2$, Left can only lower the value to zero, while Right can only raise it to one; either player would prefer to leave the position untouched if another component offered a useful move.

This contrasts with a hot position such as $\{1\mid -1\}$, where either player benefits from moving first. The distinction matters when choosing which component of a disjunctive sum to play. Exact definitions at zero temperature depend on the adopted cooling convention; a cold numerical component need not be a losing position by itself.

In a finite impartial game with the normal-play convention and no draws, a losing position is one from which the player whose turn it is cannot force a win against best play. It is also called a P-position, since the previous player wins.

A terminal position is losing. Recursively, a position is winning if it has at least one move to a losing position, and losing if every legal move goes to a winning position. Induction on the finite remaining game tree proves this classification. For a single Nim heap, only the empty heap is losing. Infinite plays and misère play need separate conventions and can change the base case.

A Nim heap of size $n$ contains $n$ counters and permits removing any positive number, leaving a heap of any size $0\le k<n$. Under the normal-play convention, its Grundy number is $n$: the reachable values are exactly $0,\dots ,n-1$, whose mex is $n$.

For several heaps, a turn changes exactly one heap. Their combined value is the Nim-sum of the heap sizes, so heaps of sizes three and three form a losing position even though either heap alone is winning. The ability to remove arbitrarily many counters is part of the rule; a bounded-removal subtraction game has different values.

The bitwise exclusive or of a list of nonnegative integers. For heap sizes in ordinary Nim, a zero nim-sum characterizes a losing position under the normal-play convention.

The convention that a player unable to move loses. In a finite impartial game without draws, a position is winning exactly when it has a move to a losing position.