The countingbury tales : fun with mathematics by Miguel de Guzman

By Miguel de Guzman

This booklet, built from the author's sequence of lectures brought in Japan in 1995, identifies and describes present effects and matters in definite parts of computational fluid dynamics, mathematical physics and linear algebra the math of a sandwich; nim; the bridges of Konigsberg; "solitaire" confinement; the mathematician as a naturalist; 4 colors are adequate; jump frog; abridged chess; the key of the oval room

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In the 18th century a French aristocrat was put into solitary confinement in the Bastille. The solitary nobleman invented a game of solitaire to make his solitude more bearable. The game became all the rage later in the England of Queen Victoria and is now enjoying a strong resurgence. You can play it very comfortably on a chess board with 32 pennies. You can use different coins, since the color and size make no difference. un a>it£ JKa^emaiics Now place the 32 coins, one in each square except for the one shown here, like this: | O |O|O| [ oI o I o I o IoIo IoIo IoI o o [ o I o I I o I o I o" o| o I o I o I o I o| o I o Io IoI IololoI Now you may begin moving the coins.

Fig. 1 This may bring back memories from when you were little. Are you able to trace the following figures without lifting pencil from paper and without repeating any line? Could you do the same thing starting and ending at the same point? Fig. 2 Fig. 3 Une l/jridges Fig. 4 of LKbnigsSerg 33 Fig. 5 Try and try again. You're almost sure to be familiar with Figure 3 and have probably done it many times, but you'll have a hard time finishing where you started. Figure 4 is so easy to trace that unless you try to do it wrong, you can start at any point and return to the same point without repeating any arcs, covering the entire path almost effortlessly.

The path that defines the maze (the italicized line in the second figure) has arcs and bifurcation points. Since we want to travel along each arc once each way, we repeat each arc twice. Once we have done this, we clearly have a figure like the ones we have been studying in this chapter with all even vertices. This way we can travel along all of it without repeating arcs, starting at any point. Furthermore, we can do it without knowing the layout of the maze. The only thing we need is to be able to mark the arcs we have already covered by some means.

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