Alex's Adventures in Numberland - Alex Bellos [99]
From a family of Sussex sheep farmers, Dudeney started working at 13 as a clerk in the civil service in London. Bored with the job, he began to submit short stories and puzzles to various publications. Eventually he was able to devote himself to journalism full time. His wife Alice wrote bestselling romantic novels about life in rural Sussex – where, thanks to her royalties, the couple was able to live in luxury. The Dudeneys, dividing their time between the countryside and London, were part of a highbrow literary scene that included Sir Arthur Conan Doyle, the creator of Sherlock Holmes, probably the most iconic puzzle-solver in all literature.
It is thought that Dudeney and Sam Loyd first made contact in 1894, when Loyd posed a chess problem in the firm belief that no one would discover his 53-move solution. Dudeney, who was 17 years Loyd’s junior, found a solution in 50 moves. The men subsequently collaborated, but fell out when Dudeney discovered Loyd was plagiarizing his work. Dudeney despised Loyd so intensely that he equated him with the Devil.
While both Loyd and Dudeney were self-taught, Dudeney had a much finer mathematical mind. Many of his puzzles touched on deep problems – frequently predating academic interest. In 1962, for example, the mathematician Mei-Ko Kwan investigated a problem about the route a postman should take in a grid of streets so as to walk on every street and do it in the shortest route possible. Dudeney had framed – and solved – the same problem in a puzzle about a mine inspector walking through underground shafts almost 50 years earlier.
Dudeney also made unintended contributions to number theory. One of his puzzles, called Root Extraction, plays on the fact that the cube roots of the following numbers are also equal to the sum of the digits that make up those numbers:
1 = 1×1×1
1 = 1
512 = 8×8×8
8 = 5 + 1 + 2
4913 = 17×17×17
17 = 4 + 9 + 1 + 3
5832 = 18×18×18
18 = 5 + 8 + 3 + 2
17,576 = 26×26×26
26 = 1 + 7 + 5 + 7 + 6
19,683 = 27×27×27
27 = 1 + 9 + 6 + 8 + 3
Numbers with this property – and there are only six of them – are now known as Dudeney numbers. Another particular forte of Dudeney’s was the geometrical dissection, which is when a shape is cut up into pieces and reassembled into another shape, like the principle behind the tangram. Dudeney found a way of converting a square into a pentagon in six pieces. His method became a popular classic because for many years it had been thought that the minimum dissection of a square into a pentagon required seven pieces.
Dudeney also discovered a novel way to cut up a triangle and make it into a square in four pieces. And he realized that if the four pieces in his solution were hinged together, they could be assembled in a chain – such that folding one way gets you a triangle and folding the other way gets you the square. He called this the Haberdasher’s Puzzle, since the shapes look like the leftover pieces of cloth that a haberdasher might h in his shop. The puzzle thus introduced the concept of a ‘hinged dissection’, and aroused such interest that Dudeney made one out of mahogany with brass hinges and presented it in 1905 at a meeting of the Royal Society in London. The Haberdasher’s Puzzle was Dudeney’s greatest legacy and has fascinated and delighted mathematicians for more than a century.
The Haberdasher’s Puzzle.
One mind that was especially enthralled by the Haberdasher’s Puzzle belonged to the Canadian teenager Erik Demaine. Demaine, who was such a prodigy that by the time he was 20 he was already a professor at MIT, was most interested in the ‘universality’ of the problem. Is it possible, he wondered, to dissect any straight-sided shape and then hinge the pieces together in a chain so it can be folded into any other straight-sided shape of equal area? He spent ten years working on the problem and in March 2008, aged 27, announced the solution to