# A Modern Treatment of the 15 Puzzle

A Modern Treatment of the 15 Puzzle Aaron F. Archer 1. INTRODUCTION. In the 1870's the impish puzzlemaker Sam Loyd caused quite a stir in the United States, Britain, and Europe with his now-famous 15-puzzle. In its original form, the puzzle consists of fifteen square blocks num­ bered 1 through 15 but otherwise identical and a square tray large enough to accommodate 16 blocks. The 15 blocks are placed in the tray as shown in Figure 1, with the lower right corner left empty. A legal move consists of sliding a block adjacent to the empty space into the empty space. Thus, from the starting placement, block 12 or 15 may be slid into the empty space. The object of the puzzle is to use a sequence of legal moves to switch the positions of blocks 14 and 15 while returning all other blocks to their original positions. 6 7 8 9 10 11 12 13 14 15 Figure 1. The starting position for the 15-puzzle. The shaded square is left empty. Loyd writes of how he "drove the entire world crazy," and that "A prize of \$1,000, offered for the first correct solution to the problem, has never been claimed, although there are thousands of http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The American Mathematical Monthly Taylor & Francis

# A Modern Treatment of the 15 Puzzle

, Volume 106 (9): 7 – Nov 1, 1999

## A Modern Treatment of the 15 Puzzle

, Volume 106 (9): 7 – Nov 1, 1999

### Abstract

Aaron F. Archer 1. INTRODUCTION. In the 1870's the impish puzzlemaker Sam Loyd caused quite a stir in the United States, Britain, and Europe with his now-famous 15-puzzle. In its original form, the puzzle consists of fifteen square blocks num­ bered 1 through 15 but otherwise identical and a square tray large enough to accommodate 16 blocks. The 15 blocks are placed in the tray as shown in Figure 1, with the lower right corner left empty. A legal move consists of sliding a block adjacent to the empty space into the empty space. Thus, from the starting placement, block 12 or 15 may be slid into the empty space. The object of the puzzle is to use a sequence of legal moves to switch the positions of blocks 14 and 15 while returning all other blocks to their original positions. 6 7 8 9 10 11 12 13 14 15 Figure 1. The starting position for the 15-puzzle. The shaded square is left empty. Loyd writes of how he "drove the entire world crazy," and that "A prize of \$1,000, offered for the first correct solution to the problem, has never been claimed, although there are thousands of

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# References (13)

Publisher
Taylor & Francis
ISSN
1930-0972
eISSN
0002-9890
DOI
10.1080/00029890.1999.12005124
Publisher site
See Article on Publisher Site

### Abstract

Aaron F. Archer 1. INTRODUCTION. In the 1870's the impish puzzlemaker Sam Loyd caused quite a stir in the United States, Britain, and Europe with his now-famous 15-puzzle. In its original form, the puzzle consists of fifteen square blocks num­ bered 1 through 15 but otherwise identical and a square tray large enough to accommodate 16 blocks. The 15 blocks are placed in the tray as shown in Figure 1, with the lower right corner left empty. A legal move consists of sliding a block adjacent to the empty space into the empty space. Thus, from the starting placement, block 12 or 15 may be slid into the empty space. The object of the puzzle is to use a sequence of legal moves to switch the positions of blocks 14 and 15 while returning all other blocks to their original positions. 6 7 8 9 10 11 12 13 14 15 Figure 1. The starting position for the 15-puzzle. The shaded square is left empty. Loyd writes of how he "drove the entire world crazy," and that "A prize of \$1,000, offered for the first correct solution to the problem, has never been claimed, although there are thousands of

### Journal

The American Mathematical MonthlyTaylor & Francis

Published: Nov 1, 1999