Alex's Adventures in Numberland - Alex Bellos [96]
The stomachion , which is also known as the ‘loculus of Archimedes’.
Since the time of Archimedes a keen interest in recreational puzzles has been a trait shared by many mathematicians. ‘Man is never more ingenious than in the invention of games,’ said Gottfried Leibniz, for instance, whose love of peg solitaire resonated with his obsession with binary numbers: a hole either has a peg in it or it doesn’t, it is either a 1 or a 0. However, the most playful of the great mathematicians was Leonhard Euler, who, in order to crack an eighteenth-century brainteaser, invented a whole new branch of mathematics.
In Königsberg, the former Prussian capital that is now the Russian city of Kaliningrad, there used to be seven bridges that crossed the River Pregel. Locals wanted to know if it would be possible to make a journey across all seven bridges without crossing any bridge more than once.
To come up with a proof that a circuit of this kind was impossible, Euler created a graph in which each landmass was represented by a dot, or node, and each bridge by a line, or link. He worked out a theorem that related the number of links touching each node to whether it was possible to make a circuit of the graph, and in this case it was impossible.
The conceptual leap Euler made was to realize that what was important to solve the problem was not information about the exact position of the bridges, but how they were connected. The London Underground map borrows this idea: it is not geographically accurate, but is faithful to how the Tube lines are linked together. Euler’s theorem launched graph theory, and presaged the development of topology, a very rich area of maths that studies the properties of objects that do not change when the object is squeezed, twisted or stretched.
Königsberg in the eighteenth century: as a map and as a graph.
Fascination in 1817 with tangrams was nothing compared with the extraordinary levels of excitement generated by the world’s second international puzzle craze. From the day in December 1879 when the Fifteen puzzle launched in a Boston toyshop, manufacturers couldn’t meet the demand. ‘Neither the wrinkled front of age nor the cherubic brow of childhood is proof against the contagion,’ declared the Boston Post.
The Fifteen puzzle consisted of 15 square wooden blocks placed in a square carton so that the blocks made a × 4 square with one square missing. The blocks were numbered 1 to 15 and put in the box randomly. The aim of the puzzle was to slide the blocks around the 4 × 4 square using the empty space and finish with them positioned in numerical order. Playing with the Fifteen puzzle was so addictive and fun that the fad soon spread from Massachusetts to New York, and then across the United States. ‘It has swept over the land from East to West with the violence of the sirocco, scorching men’s brains as it passed, and apparently making them temporarily insane,’ quivered the Chicago Tribune. According to the New York Times, no pestilence ‘has ever visited this or any other country which has spread with [such] awful celerity’.
The puzzle soon went overseas, with one shop in London reportedly selling nothing else. Within six months it had reached the other side of the world. ‘Not a few have already been driven insane,’ claimed a letter in the Otago Witness, New Zealand, on 1 May 1880.
The Fifteen puzzle was initially known as the Gem puzzle.
The Fifteen puzzle was the creation of Noyes Chapman, a postmaster in upstate New York, who almost two decades previously had been trying to make a