By R.; Hilbert, D. Courant
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Additional resources for Methods of Mathematical Physics (2 Volumes)
Similar notational systems for numbers can be set up using any number as base. The most important of these alternative notational systems, called binary, uses base 2. Now there are only two digits, 0 and 1, and the value of a digit doubles at every step from right to left. In binary, 10 means 2 (in our usual decimal notation), 100 means 4, 1000 means 8, 10000 means 16, and so on. To get numbers that are not powers of 2, we add distinct powers of 2 together. For instance, 23 in decimal is equal to 16 þ 4 þ 2 þ 1 which uses one 16, no 8, one 4 one 2, and one 1.
Various versions of this method were found by several mathematicians, who kept their methods secret. Eventually Girolamo Cardano, aka Jerome Cardan, published them in one of the world’s great algebra books, the Ars Magna (The Great Art). When he did, 40 Incredible numbers one of the others accused him of stealing his secret. It wasn’t that unlikely. Around 1520, Cardano was broke. He turned to gambling as a source of ﬁnance, exploiting his mathematical abilities to improve his chances of winning.
The original position, with just one pair of blocks (14 and 15) swapped, is an odd permutation of the required ﬁnal position. But the requirement that the empty square ends up back where it started implies that legal moves lead only to even permutations. Therefore legal moves, starting from any given initial state, can reach exactly half of the 16! possible rearrangements, which is 10,461,394,944,000 arrangements. By trial and error it is impossible to explore more than a fraction of these possible arrangements, which can easily persuade people that if only they keep trying, they might just stumble across a solution.
Methods of Mathematical Physics (2 Volumes) by R.; Hilbert, D. Courant