A Variant of Game of Sliding Coins (2504.18606v1)

Abstract: Here, we present a variant of the sliding coins game. Two coins are placed on distinct squares of a semi-infinite linear board with squares numbered $0, 1, 2, dots, $. Two players take turns and move a coin to a lower unoccupied square. When a coin is pushed to the outside of the linear board, the players cannot use this coin anymore. In this game, we have another operation of moving coins: moving the coin on the right leftward and pushing the coin on the left. This last operation complicates this game's mathematical structure, but we have managed to make formulas for Grundy numbers. Since all the positions of this game of two coins are the next player's winning position, this game of two coins is trivial as a game. However, by making the sum of two games, we get a meaningful game in which the player who plays for the last time is the winner. With these Grundy numbers formulae, we get the winning strategy.