Wednesday, 13 May 2009

The Fifteen Puzzle



Before there was Sudoku, before there was Rubik's Cube, there was the fifteen puzzle. In 1880 it was THE hot game to play, and seemingly everyone did, "About the year 1880, everyone in Europe and America was engaged in the solution... ". The image above shows an old copy that was for sale somewhere (sorry, I should have made a note).
The object of the game was to slide the fifteen squares into the open space and by doing so put them in correct order. There are 15! or 1,307,674,368,000 ways to put the fifteen squares into the box, and exactly half of them are impossible to solve.

Last night as I was thumbing through my 1917 copy of H. E. Licks, "Recreations in Mathematics", and while glancing through the section on the fifteen puzzle came across the following: "It has been stated that this interesting puzzle was invented in 1878 by a deaf and dumb man as a solitaire game. "

Ok, maybe, but it just seemed too improbable.... Occam's Razor and all that... besides, I thought I had heard that it was a Sam Loyd puzzle. A litte quick research and I found a reveiw of The 15 Puzzle: How It Drove the World Crazy by Jerry Slocum and Dic Sonneveld at MAA online.

If you are interested in the actual book, you can order here; and support my beautiful wife's charitable foundation.


Sam Loyd was almost certainly the premier puzzle master of the late Nineteenth Century, and he was not shy about claiming credit for almost anything, and did claim the invention of the Fifteen Puzzle in his books. The more likely truth, is that , "The puzzle was "invented" by Noyes Palmer Chapman, a postmaster in Canastota, New York, who is said to have shown friends, as early as 1874, a precursor puzzle consisting of 16 numbered blocks that were to be put together in rows of four, each summing to 34. Copies of the improved Fifteen Puzzle made their way to Syracuse, New York by way of Noyes' son, Frank, and from there, via sundry connections, to Watch Hill, RI, and finally to Hartford (Connecticut), where students in the American School for the Deaf started manufacturing the puzzle ((So there was the connection to deaf and dumb inventor story) and, by December 1879, selling them both locally and in Boston (Massachusetts). Shown one of these, Matthias Rice, who ran a fancy woodworking business in Boston, started manufacturing the puzzle sometime in December 1879 and convinced a "Yankee Notions" fancy goods dealer to sell them under the name of "Gem Puzzle". In late-January 1880, Dr. Charles Pevey, a dentist in Worcester, Massachusetts, garnered some attention by offering a cash reward for a solution to the Fifteen Puzzle. The game became a craze in the U.S. in February 1880, Canada in March, Europe in April, but that craze had pretty much dissipated by July. Apparently the puzzle was not introduced to Japan until 1889. Noyes Chapman had applied for a patent on his "Block Solitaire Puzzle" on February 21, 1880. However, that patent was rejected, likely because it was not sufficiently different from the August 20, 1878 "Puzzle-Blocks" patent (US 207124) granted to Ernest U. Kinsey."(from Wikipedia)... ( Kinsey's patent has an application date in November of 1877, and includes interlocking pieces so they stayed in the box like modern fifteen puzzles do An image from the patent is here.

What Sam Loyd did do, was realize that in 1/2 of the 15! possible ways to put the 15 squares in the puzzle, it could not be solved. He immediatly offered a $1000 prize to anyone who could solve it in his newspaper aritcle, magazines, and books, and draw a ton of free advertising.

Eventually it became known that the puzzle could not always be solved, and some of the interest waned, but the puzzle remained popular enough that it was a subject for methaphor for writers. "But Big Jack Fish Lake was two days' travel away, and meanwhile my ankle made life intolerable, and the map proved more maddening than the fifteen puzzle.", from On Snow-Shoes to the Barren Ground, by Caspar Whitney ; 1896 - page 72)

I put a copy of the puzzle on my side bar for those who wish to give it a try, or if you want, you can play a version that has 8 numbers (3x3 board) 15, or 24 numbers (5x5 board) . One of the things that was discovered about the Loyd Puzzle, the one that could NOT be done, was that if you left the blank in the top left hand corner (putting 1-2-3 in the top row, then 4-5-6-7 next, etc), it would be possible to put all the numbers in the correct order. This would be the same on the solvable puzzle as getting all the numbers in the right order except reversing the 14 and 15... try it.

Here is an additional "15 Puzzle" not related to the physical puzzle above.  I found it on Greg Ross' Futility Closet a while back:
A problem from the 1999 Russian mathematical olympiad:
Show that the numbers from 1 to 15 can’t be divided into a group A of 13 numbers and a group B of 2 numbers so that the sum of the numbers in A equals the product of the numbers in B.
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