Things to Make and Do in the Fourth Dimension: A Mathematician's Journey Through Narcissistic Numbers, Optimal Dating Algorithms, at Least Two Kinds of Infinity, and More by Matt Parker

Author:Matt Parker [Parker, Matt]
Language: eng
Format: epub, azw3
ISBN: 9780374710378
Publisher: Farrar, Straus and Giroux
Published: 2014-12-02T00:00:00+00:00

The Tower of Hanoi from Récréations mathématiques.

The Tower of Hanoi, or to quote Lucas exactly, ‘La Tour d’Hanoï’, takes up only three pages in the book, roughly a third of which is dedicated to a large diagram of the game set-up – a large diagram which is no longer in copyright, so I’ve reproduced it here for you. As I do with all my now-dead maths heroes, I went hunting for some of Lucas’s writings so I could get a feel for him as a person. Fortunately, Récréations mathématiques is available online as a free ebook which I could easily download. Unfortunately, as I mentioned before, it is of course entirely in French. So I emailed a copy of the three Hanoi pages to my French-teacher brother-in-law to translate.

When he emailed me his translation, I was surprised to see that half the text isn’t about the Tower of Hanoi at all, but rather something called the Wheat and Chessboard Problem. This is a challenge to calculate the total number of grains of wheat on a chessboard if you put one in the first square, two in the second, four in the third, and so on, doubling the amount for each of the sixty-four squares. (hint.subtlety = ‘blatant’; there is a link to the Tower of Hanoi.) In the first section of text, Lucas talks about the Tower of Hanoi toy produced around 1883 by a Professor N. Claus (from Siam). Curiously, the original instructions which came with the toy, written by Claus, said that more details about the game were available in the book Récréations mathématiques by Lucas. It all seems very circular and incestuous until you realize that ‘Claus’ is an anagram of ‘Lucas’, and the full ‘N. Claus (de Siam)’ is an anagram of ‘Lucas d’Amiens’ (Lucas was educated in Amiens).3 It seems Lucas and his pseudonym are having a great time patting each other’s backs.

Back to the game. The two-disc version is nice and straightforward. We’ll call the small disc A and the bigger one under it B. Move A on to the holding pole, then shift B to the target pole, before moving A back on top of it. Three easy moves. We could even write them ‘A B A’ to show the order. If you start with three discs, you can move the whole tower in seven moves, and a four-disc tower can be relocated in only fifteen disc movements. Find some different-sized discs and have a play around, slowly adding more until you’re happy you could relocate a five-disc tower. Most toy versions of the game are sold with around seven to ten discs, which would result in a fairly swift end to the world. Thankfully, the priests of legend have to move a 64-disc tower before the world ends, which buys us all a bit more time.

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