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. 


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 immediately offered a $1000 prize to anyone who could solve it in his newspaper article, 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 metaphor 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)

Ok, there is sort of a way to solve the 15 puzzle after the 14,15 swap   i think the book has 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.

There is another Fifteen puzzle that has been very popular in the past, the Fifteen Peg Puzzle

Fifteen Peg Puzzle

The puzzle shown above, often called a triangular peg solitaire game is so common you may have last seen it on your table at Cracker Barrel Restaurants.
Below I will give a solution to a common form of the puzzle, so don't look too far below if you want to test out your skills without clues or a solution. If you want to play the online version of the game, try your luck here. 
If you get really interested, you can try games removing any one of the pegs instead of a corner (which is NOT the easiest possible solution).    If you get frustrated, here are some good hints about the game from an excellent page by George Bell.

Note the symmetry of the triangular board: there are three corner holes, three interior holes , and three holes at the midpoint of each edge , plus six "other" holes .
The following rules of thumb are based on a mathematical analysis of the game and should help you solve the puzzle

1. Avoid jumping into a corner. Of course, in some situations (such as beginning without a corner peg) this is the only jump possible.
2. Avoid any jump which starts from one of the  interior holes. Such a move is almost always a dead end (none of the solutions include this jump).
3. The easiest place to begin the game is with the missing peg (hole) at one of the  midpoint locations. The hardest place to begin is with the missing peg at one of the  interior holes. 

Complete solution below:::

The truth is, there are thousands of possible solutions to the game. The solution I just came across is for the case in which you start with the top corner (the one hole) empty, and finish with a single peg in that hole (which I just learned is called a single vacancy complement solution). If you number the holes starting at an apex of the triangle with one, then two and three on the next row, and continue with 11 through 15 on the bottom row, then moves can be described with (x,y) coordinates where the term (x,y) means move the peg in hole x to hole y.
A winning solution to the 15-hole triangular peg solitaire game using this method is: (4,1), (6,4), (15,6), (3,10), (13,6), (11,13), (14,12), (12,5), (10,3), (7,2), (1,4), (4,6), (6,1).

Not only does this solution leave the final peg in the original empty hole, but the sum of all the x,y hole numbers in the solution is prime (179) and if I time this right it should be first posted on the 179th day of the year 2012. By the way, if you only sum the values of the landing holes, that's prime also (73).

I first came across this curious little fact at the Prime Curios page.

Haven't pursued it yet, but since there are thousands of solutions to this puzzle, I assume that there would be other sequences which produced a prime also.