A Variant of Game of Sliding Coins
Ryohei Miyadera, Hikaru Manabe, Unchon Lee
TL;DR
The paper introduces a variant of sliding coins on a semi-infinite board and analyzes its impartial game structure using Grundy theory. It develops explicit Grundy-number formulas via the Sprague-Grundy framework, including the classifications $G_{n,1}$, $G_{n,2}$, $G_{n,3}$ for residues modulo $6$, and analyzes inter-set moves to identify $\\mathcal{P}$-positions. Although the single two-coin game is trivial in isolation, the authors show that the disjunctive sum of two such games becomes meaningful, enabling a complete winning strategy through nim-sum computations. This work extends combinatorial game theory for sliding-coin variants and demonstrates how component games can be combined to yield nontrivial impartial games with clear winning strategies.
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.