The 15 puzzle: how it drove the world crazy

Some years ago, I wrote an article about the 15 puzzle that began, “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” [1]. I’ve always been pleased with myself for managing to slip the word “impish” into a published mathematics article. As it turns out, this devilish descriptor was the most accurate part of that sentence, as Slocum and Sonneveld document in their new book The 15 Puzzle: How it Drove the World Crazy. The 15 puzzle did once cause an intense craze that spread like wildfire across America and overseas, and it is indeed famous to this day. However, the initial fad did not occur until 1880, and Sam Loyd had nothing to do with it until eleven years later, when he started to claim in print that he had invented the puzzle. Through meticulous research using primary sources, Slocum and Sonneveld not only expose Sam Loyd’s fraudulent claims, but also argue convincingly that the actual inventor was Noyes Chapman, the postmaster of Canastota, New York. The 15 puzzle is a sliding block puzzle consisting of 15 numbered square blocks placed inside a frame large enough to accommodate 16 blocks in a 4 by 4 grid (see Figure ***). The empty space allows the solver to slide any of the adjacent blocks into the space. Given a starting configuration, the puzzle is to reach some specified target configuration via a sequence of such moves. The instructions written on the cover of the original puzzle read, “Place the Blocks in the Box irregularly, then move until in regular order” (p. 8). It did not specify exactly what is meant by “regular order,” but it did contain a picture of the blocks arranged as shown in Figure ***. Later posers of the puzzle were more careful to explicitly state this as the target state. For the purposes of our discussion, let us call this the canonical state. Early fans of the puzzle discovered that they could often reach a configuration that differed from the canonical state only in that the 14 and 15 were swapped, but try as they might, they could not complete the puzzle. Thus, the problem of solving the puzzle from this starting state became the standard challenge, which we will call the 14-15 puzzle. Beginning in 1880, numerous cash prizes of up to $1000 American (a princely sum in those days) were offered for the solution of the 14-15 puzzle. There have been numerous published accounts of people spending endless hours engrossed in the puzzle, but nobody ever successfully claimed these prizes. An intriguing mathematical fact about the 15 puzzle is that for exactly half of the 16! possible initial configurations, the puzzle is impossible to solve. It should come as no surprise that the 14-15 puzzle starts from one of these impossible configurations. The set of solvable configurations can be easily described using the theory of even and odd permutations. First, it should be clear that the set of all solvable configurations are precisely those that can be reached starting from the canonical state, since each move is reversible. Second, let us imagine the blank space to contain a block that we will call the “blank block.” Then each move consists of swapping the blank with one of the blocks that is adjacent to it horizontally or vertically.