The following game is well-known among students of mathematics, and it is easy to understand for anyone else as well:—Two players take turns in putting down identical round coins on a round table. The coins must not overlap of course, only their edges can touch each other. No coin already placed can be moved later. As the area of a table is finite, sooner or later we get to the point when no more coins can be placed on the table this way. The player who managed to place the last coin wins. Well, who wins if both sides play perfectly: the first player or the second? Compared to chess, the game is not too interesting, but it illustrates a mathematical principle very well. At first, one would think that the answer must depend on the relative size of the table and the coins. However, even if the two sizes are given, it is not so easy to solve the problem, except if the table happens to be actually smaller than the coin. If the table can hold only one coin, the first player evidently wins. So we can’t say that ’Black’ is always OK in this game! He can never win at a small table. Our second thought could be that at a big table, the situation is extremely complicated, requiring a hopelessly great number of geometrical calculations. We suddenly feel that the game is as complex as chess. However, there is a brilliant idea that helps: symmetry! The concept of symmetry often helps to prove things in mathematics and other branches of the natural sciences. So what can we do with symmetry in this case? If the second player could imitate the first player using one of the two symmetries of the table (axial or central), he could win. This does not work, however, as the two coins may overlap. It is actually the first player who can use the ’aping’ strategy! He puts down the first coin in a way that its centre is identical with the centre of the table. This move can be made on any ’board’, no matter how small it is. Then ’Black’ puts down a coin somewhere if he can. And now the first player places his second coin using central symmetry. He can do that! Then it’s ’Black’s’ turn again, and ’White’ can follow his simple strategy. He can always do so, as the coins are always placed in a centrally symmetrical way after his moves. As long as Black still has a move, White has one too. Only White can win with this strategy, as the moment comes sooner or later when Black runs out of space! If we could find a similar line of argumentation in chess as well, we could justify White’s superiority. However, no one has ever had such an idea! Read More ➤
Posted on: Sat, 10 Aug 2013 11:59:38 +0000
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