Abstract
Finitely many coins are placed on distinct squares of a semi-infinite linear board ruled into squares numbered 0, 1, 2,… Each of the two players alternately moves a coin to a lower unoccupied square, at most k squares from its present position, where k is an arbitrary fixed positive integer. The player first unable to move loses and his opponent wins. We give complete solutions for the case of two coins for any k; and for any number of coins when k = 2 m − 1 ( m ⩾ 1). We also prove that, for any k, the Sprague-Grundy function of the game is invariant under translations of coins by k + 1.
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