Coin-Moving Puzzles

TL;DR

This paper introduces a new class of coin-moving puzzles, characterizes their solvability on specific grids, and provides efficient algorithms for solving them, advancing understanding of these combinatorial games.

Contribution

It precisely characterizes which coin-moving puzzles are solvable on square and triangle grids using a fixed number of extra coins, with accompanying polynomial algorithms.

Findings

Characterization of solvable puzzles on square and triangle grids.

Polynomial-time algorithms for solving these puzzles.

Introduction of the concept of extra coins for solvability.

Abstract

We introduce a new family of one-player games, involving the movement of coins from one configuration to another. Moves are restricted so that a coin can be placed only in a position that is adjacent to at least two other coins. The goal of this paper is to specify exactly which of these games are solvable. By introducing the notion of a constant number of extra coins, we give tight theorems characterizing solvable puzzles on the square grid and equilateral-triangle grid. These existence results are supplemented by polynomial-time algorithms for finding a solution.