The Sam Loyd 15-Puzzle

Abstract

This report presents an approach to solve Sam Loyd?s famous 15-puzzle. The 15 or sliding puzzle is traditionally represented as a 4 * 4 board with tiles numbered from 1 to 15 arranged in numerical order from top left to bottom right of the board. The object of the puzzle is to rearrange the tiles in this order from any solvable scrambled starting position. A solvable board is one where the start configuration is an even permutation of the solved board. This report describes an algorithm that generates solutions to solvable boards not only for the 15 puzzle, but also for boards of different dimensions. It adapts a divide and conquer approach outlined in Ian Parberry?s paper (1997). Firstly, a brief discussion on the definition and background of the puzzle is offered. An in-depth explanation of what constitutes a solvable puzzle board is presented - an analysis of even permutations, which is fundamental to generating a solvable board, is given. From this, an algorithm to check a permutation for evenness is then illustrated. The adapted algorithm to solve the 15 or (n2-1) puzzle is then presented along with the theory and reasoning behind the adaptation, demonstrating different solution cases depending on where a tile is situated on the puzzle board. Following this, a discussion on special board configurations that cannot be directly solved by the puzzle algorithm is offered. A second report will follow on the steps required to generate the worst case number of moves of a board of a given dimension, which demonstrates that the adapted algorithm solves the worst case board in substantially less moves than the one described in [Parberry 1997]. In the appendix of this report appears each series of moves used in the algorithm to solve a puzzle board.