Coin-Moving Puzzles

Summary

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 characterizingsolvable puzzles on the square grid and equilateral-triangle grid. These existence results are supplemented by polynomial-time algorithms for finding a solution.

1. Introduction

Consider a configuration of coins such as the one on the left of Figure 1. The player is allowed to move any coin to a position that is determined rigidly by incidences to other coins. In other words, a coin can be moved to any position adjacent to at least two other coins. The puzzle or 1-player game is to reach the configuration on the right of Figure 1 by a sequence of such moves. This particular puzzle is most interesting when each move is restricted to slide a coin in the plane without overlapping other coins.

This puzzle is described in Gardner's Mathematical Games article on Penny Puzzles [7], in Winning Ways [1], in Tokyo Puzzles [6], in Moscow Puzzles [8], and in The Penguin Book of Curious and Interesting Puzzles [11]. Langman [9] shows all 24 ways to solve the puzzle in three moves. Another classic puzzle of this sort [2; 6; 7; 11] is shown in Figure 2. A final classic puzzle that originally motivated our work is shown in Figure 3; its source is unknown. Other related puzzles are presented by Dudeney [5], Fujimura [6], and Brooke [4].

The preceding puzzles always move the centers of coins to vertices of the equilateral-triangle grid. Another type of puzzle is to move coins on the square grid, which appears less often in the literature but has significantly more structure and can be more difficult.