Variations of the 15 Puzzle

VARIATIONS OF THE 15 PUZZLE A thesis submitted to the Kent State University Honors College in partial fulfillment of the requirements for University Honors by Lisa Rose Hendrixson May, 2011 Thesis written by Lisa Rose Hendrixson Approved by , Advisor , Chair, Department of Mathematics Accepted by , Dean, Honors College ii TABLE OF CONTENTS LIST OF FIGURES. .iv ACKNOWLEDGEMENTS . v CHAPTERS I INTRODUCTION . 1 II THE HISTORY OF THE 15 PUZZLE . 3 III MATHEMATICS AND THE 15 PUZZLE . 6 IV VARIATIONS OF THE 15 PUZZLE . 14 V CONCLUSION . 22 BIBLIOGRAPHY . 23 iii LIST OF FIGURES • Figure 1.The 15 Puzzle . 2 • Figure 2.Pictoral representation of the above permutation. 8 • Figure 3.The permutation multiplied by the transposition (7; 8) ..................9 • Figure 4.Two odd length cycles inside the 15 Puzzle . 10 • Figure 5.The two cycles, put together. 11 • Figure 6.Permutting 3 blocks cyclically. 12 • Figure 7.Bipartite graph of the 15 Puzzle. 13 • Figure 8.Starting position for the first variation. 15 • Figure 9.Creating a single transposition inside the variation. 16 • Figure 10.Showing the switch of the blank spaces. 16 • Figure 11.Puzzle with a fixed block. 17 • Figure 12.Bipartite graph of a puzzle wth a glued-down block. 21 iv ACKNOWLEDGEMENTS I would like to thank Dr. Donald White for all his help and support during the process of writing this thesis. Without his dedication, it would not have been possible. Also, I would like to thank Dr. Mark Lewis, Dr. Elizabeth Mann, and Dr. Sara Newman for their willingness to serve on my defense committee and for all of their helpful comments and support along the way. v INTRODUCTION The 15 Puzzle: to look at, it does not look like much. Just a tray, with 15 little numbered blocks. Admittedly, these blocks come out of the tray easily, but this is only exciting for as long as it takes to find the piece that fell out. It is difficult to imagine that such a thing would excite masses of people to a fervor equivalent to that of a group of mid- dle school girls at a boyband concert. Who would have guessed the mathematical insight lurking beneath the simple design? The answer is, perhaps, too simple: a mathematician would indeed find the hidden complexities. The 15 Puzzle is a device that few know much about in the present time. It has lost its intrigue to newer and more complicated puzzles that have taken center stage as both toys and catalysts for mathematical thought. Most people fail to understand what a 15 Puzzle is without a somewhat longwinded explanation. Essentially, the 15 Puzzle is a small tray with 15 numbered blocks that can be pulled out of the tray and rearranged in any manner that the player desires. The desired position is when the blocks are arranged as in Figure 1, with the 1 block in the top left corner and the rest of the blocks continuing in numerical order left to right and then top to bottom. From there, the blocks can be removed from the tray and replaced in any order the player wants. It should be stated that taking the pieces out and putting them back in the solved position is cheating. The goal of this thesis is to educate the reader about the 15 Puzzle, including its 1 2 definition, the somewhat murky history, and the mathematics behind it. Figure 1: The 15 Puzzle CHAPTER ONE THE HISTORY OF A PHENOMENON The origins of the 15 Puzzle are shrowded in mystery. The most popular story is that the 15 Puzzle was invented by Sam Loyd, an American in the mid 19th Century. Loyd was a famous and prolific puzzle maker, who, sadly, was particularly lax when it came to giving credit to actual inventors. Regardless, he became known as “America’s Greatest Puzzlist,” with some help from the work of Henry Dudeney, his British counterpart [12]. Loyd began playing chess at the age of 14, and within two years became famous for related problems. This led to a job as Problem Editor for the American Chess Journal and the writing of a book entitled Chess Strategy. However, this book has a copyright de- screpancy: while the date says 1878, Alain White wrote in his book about Loyd that Chess Strategy was not published until 1881. If this is to be believed, then Loyd would have been working on the book while the 15 Puzzle craze was going on, a not impossible feat, but perhaps improbable. At any rate, Loyd’s name was never mentioned in connection with the 15 Puzzle during the puzzle’s peak of popularity. Instead, Loyd first mentioned having invented the puzzle a decade after the craze had ended. After his death, Loyd’s son, Sam Loyd, Jr., took up his father’s business and even the credit for some of his father’s puzzles. What he did not do, however, is give his father, or himself, any credit concerning the 15 Puzzle [12]. 3 4 Another theory on the origin of the 15 Puzzle is that a child in a deaf-mute school in Hartford, Connecticut invented the puzzle. No evidence to support this claim has ever been found [12]. A third theory about the invention of the 15 Puzzle is that Noyes Chapman developed the puzzle in 1879. Chapman was a New York postmaster. Chapman did apply for a patent for a puzzle he called “Block Solitare Puzzle.” However, the application was re- jected because of the similarity to a puzzle developed more than a year prior by Ernest U. Kinsey. Kinsey had a 6 × 6 puzzle with lettered blocks, instead of numbered ones. Despite this, it seems that Chapman gave puzzles to Anna and James Belden. James Belden was an ex-mayor of Syracuse, New York and a business associate of Noyes Chapman’s son, Frank Chapman. From there, Mr. and Mrs. Belden took the puzzle to the community of Watch Hill, Rhode Island, where copies were manufactured based on the originals from Noyes Chapman. It is unknown how the puzzle made its way from Watch Hill to Hartford, or further still to Boston, where Matthias Rice began manufacturing and selling the puzzle in earnest [12]. Rice began making a verson of the puzzle he referred to as “The Gem Puzzle” in December 1879 in his woodworking shop in Boston, Massachusetts. He sold the puzzles for 75 cents, and the idea was picked up by other manufacturers when the puzzle proved to be popular. The craze spread outward, west to the rest of America and east into Europe, Asia, and Australia, where it remained popular until as late as June 1880. It was hypothe- sized that players would go mad from working the puzzle, although doctors discredit that 5 notion as those who went insane had demonstrated evidence of the condition prior to expo- sure to the 15 Puzzle. The trouble, of course, stemmed from the fact that the puzzle is not always solvable, an assumption that the average player made. Rewards were offered for a solution to the problem of switching the 14 block and the 15 block. It took a rigorous proof using group theory to show that these people need not have bothered: switching the 14 and 15 blocks creates a puzzle which is unsolvable. This proof and another, more recent proof will be demonstrated in the next chapter [12]. The 15 Puzzle opened up a market for similar puzzles, many with letters, or dif- ferent dimensions, etc. A three-dimensional puzzle was even constructed and patented by Charles I. Rice in 1889. The 15 Puzzle left its impact on the world, making way for the Rubik’s Cube craze of the 1980s. But for those who prefer a simple pastime that is easily fit into one’s pocket, the 15 Puzzle is still available for sale. For the technologically savvy, computer programs and internet sites offer the puzzle, although only in solvable configura- tions. And, naturally, mathematicians, being an obsessive lot, still play with the puzzle and the mathematical thoughts it provokes [12]. CHAPTER TWO MATHEMATICS AND THE 15 PUZZLE It should come as little surprise that, while the general population obsessed over the 15 Puzzle, so too did mathematicians. The puzzle is solved when the blocks are in numerical order, left to right, and top to bottom. The first proofs of possible (solvable) positions and impossible (unsolvable) positions appeared in early 1880. The proof most often stated as the first is that of William Woolsey Johnson concerning the fact that the odd permutations are impossible. A permutation is the product of transpositions. A transposition is when two blocks, from anywhere in the puzzle are switched, while leaving the rest alone, often by taking those two blocks straight up out of the tray and switching them. Johnson’s article was published in the December 1879 issue of the American Journal of Mathematics [14], but was not actually released until April 1880 because the journal was running late at the time. The American Journal of Mathematics published Johnson’s proof alongside another proof by William E. Storey, who proved that all even permutations are solvable positions. Even permutations are those that are the product of an even number of transpositions, and odd permutations are those that are the product of an odd number of transpositions [12]. In general, mathematics from the 1800s is difficult to read and understand. How- ever, Jerry Slocum and Dic Sonneveld have given a brief sketch of Johnson’s proof in their book entitled The 15 Puzzle:How It Drove the World Crazy [12].

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