Variations on a Theme of Euclid

The game of Euclid is an impartial game played between two players. A position in the game is a pair of integers (a, b). A move consists of replacing the current position with one in which the larger of a and b has been reduced by any multiple of the smaller. The game ends when the two numbers are equal. The players alternate moves, and the winner is the last player to make a move. Several variations take the form of restrictions on the moves available to the players. One important class of restrictions takes the form of a set Λ of positive integers from which the number of multiples a player removes on a turn must be chosen. Of particular interest are versions with ”dynamic” restrictions. In these variations of Euclid, the maximum multiple which can be removed on a turn is governed by some given function. In this way, the set of available moves changes as the game proceeds. It is shown how all versions considered can be recast as sequential take-away games, and this transformation is frequently used to find winning strategies.