Download Across the Board: The Mathematics of Chessboard Problems by John J. Watkins PDF

By John J. Watkins

Around the Board is the definitive paintings on chessboard difficulties. it isn't easily approximately chess however the chessboard itself--that easy grid of squares so universal to video games worldwide. And, extra importantly, the attention-grabbing arithmetic at the back of it. From the Knight's travel challenge and Queens Domination to their many diversifications, John Watkins surveys all of the recognized difficulties during this unusually fertile region of leisure arithmetic. Can a knight stick with a course that covers each sq. as soon as, finishing at the beginning sq.? what percentage queens are wanted in order that each sq. is concentrated or occupied via one of many queens?

Each major subject is taken care of intensive from its old perception via to its prestige this day. Many attractive options have emerged for uncomplicated chessboard difficulties in view that mathematicians first started engaged on them in earnest over 3 centuries in the past, yet such difficulties, together with these related to polyominoes, have now been prolonged to three-d chessboards or even chessboards on strange surfaces reminiscent of toruses (the similar of enjoying chess on a doughnut) and cylinders. utilizing the hugely visible language of graph thought, Watkins lightly publications the reader to the vanguard of present study in arithmetic. by means of fixing many of the many workouts sprinkled all through, the reader can proportion totally within the pleasure of discovery.

Showing that chess puzzles are the start line for vital mathematical rules that experience resonated for hundreds of years, around the Board will captivate scholars and teachers, mathematicians, chess fanatics, and puzzle devotees.

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Extra info for Across the Board: The Mathematics of Chessboard Problems

Example text

6 Take all four aces, all four kings, all four queens, and all four jacks from a single deck of playing cards. Arrange these sixteen cards in four rows and four columns so that each row and each column contains all four card values as well as all four suits. If you succeed in doing this, you will have created a Graeco-Latin square of order 4. Martin Gardner even tossed in an extra challenge: arrange the cards so that the two diagonals of the square also satisfy this same condition. 14 we first draw the knights graph in its natural position on the chessboard and also at the same time number the squares of the board in a convenient fashion.

This example was first shown to me by a student, David Carlson, who came across it in W. W. Rouse Ball’s Mathematical Recreations and Essays [2]. It serves to illustrate the industriousness with which one of the greatest mathematical minds of all time attacked the Knight’s Tour Problem. In this example Euler provides us with a powerful and flexible technique that will prove useful to us later over and over again in a variety of situations. 2 that misses only four squares: a, b, c, d. Rather than backtracking at this point and trying different paths either at random or one by one, since neither approach seems at all promising, Euler’s strategy is to deal with what is missing from our current tour methodically one step at a time.

Note that a knight at a square colored A can only move to a square colored C, so when a knight visits a square colored A while traveling along the knight’s tour, the knight must have just come from a square colored C and be just about to go to still another square colored C next on the tour. Moreover, since there are an equal number of squares colored A and squares colored C, when a knight moves from a square colored A to a square colored C, it can’t then go on from the square colored C to a square colored D, for the knight would still have to eventually return to another square colored C before being able to get back to a square colored A.

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