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A chess engine is a computer program that can play the game of chess, it can also refer not just to a program, but to a whole hardware machine. See also computer chess. However, usually the term chess engine is used to refer to a chess playing program that does not have its own user interface.
Most chess engines use the common approach of attempting all possible moves from a given position and then choosing the best one with the help of a search (mostly alpha beta search, which is an improvement over the minimax algorithm) and evaluate function.
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