Alex's Adventures in Numberland - Alex Bellos [97]
The Fifteen puzzle was especially tormenting to those who attempted it because sometimes it was solvable and sometimes it was not. Once the blocks were randomly put in there seemed to be only two outcomes: either they could be rearranged in numerical order, or they could be rearranged so that the first three rows were in order but the last line went 13-15-14. The craze was fuelled, in part, by a desire to work out if it was possible to get from 13-15-14 to 13-14-15. In January 1890, a few weeks after the first puzzle went on sale, a dentist in Rochester, New York, placed an ad in the local paper offering a $100 prize and a set of false teeth to anyone who could prove it either way. He believed it was impossible – but needed some help with the maths.
Bafflement with the Fifteen puzzle spread from the world’s living rooms to the halls of academe, and once the professionals got involved, the puzzle went from insanity-inducing unsolvable to satisfactorily unsolvable. In April 1890 Hermann Schubert, one of the outstanding mathematicians of his day, published in a German newspaper the earliest proof that 13-15-14 was an unsolvable position. Shortly thereafter, the recently founded American Journal of Mathematics also published a proof, confirming that half of the total of all starting positions in the Fifteen puzzle will produce a final solution of 13-14-15, and half will end up 13-15-14. The Fifteen puzzle remains the only international craze in which the puzzle does not always have a solution. No wonder it drove people mad.
Like the tangram, the Fifteen puzzle has not totally disappeared. It was the forerunner of the sliding-block puzzles that are still found in toyshops, Christmas crackers and corporate marketing gift packs. In 1974 a Hungariann was devising ways to improve the puzzle when he was struck with the idea of reinventing it in three dimensions. The man, Ernö Rubik, came up with his prototype, the Rubik’s Cube, which went on to become the most successful puzzle in history.
In his 2002 book The Puzzle Instinct, the semiotician Marcel Danesi wrote that an intuitive ability to solve puzzles is part of the human condition. When we are presented with a puzzle, he explains, our instincts drive us to find a solution until we are satisfied. From the riddle-asking sphinx of Greek mythology to the detective mystery, puzzles are a common feature across time and cultures. Danesi argues that they are a form of existential therapy, showing us that challenging questions can have precise solutions. Henry Ernest Dudeney, Britain’s greatest puzzle-compiler, described solving puzzles as basic human nature. ‘The fact is that our lives are largely spent in solving puzzles; for what is a puzzle but a perplexing question? And from our childhood upwards we are perpetually asking questions or trying to answer them.’
Puzzles are also a wonderfully concise way of conveying the ‘wow’ factor of maths. Often they require lateral thinking, or rely on counter-intuitive truths. The sense of achievement gained from solving a puzzle is an addictive pleasure; the sense of failure from not solving one almost unbearably frustrating. Publishers realized pretty quickly that fun maths had a market. Amusing and Entertaining Problems that Can Be Had with Numbers (very useful for inquisitive people of all kinds who use arithmetic) by Claude Gaspard Bachet came out in France in 1612. It included sections on magic squares, card tricks, questions in non-decimal bases, and think-of-a-number problems. Bachet was a serious scholar who translated