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

By John J. Watkins

Around the Board is the definitive paintings on chessboard difficulties. it's not easily approximately chess however the chessboard itself--that uncomplicated grid of squares so universal to video games around the globe. And, extra importantly, the interesting arithmetic at the back of it. From the Knight's journey challenge and Queens Domination to their many adaptations, John Watkins surveys the entire recognized difficulties during this strangely fertile zone of leisure arithmetic. Can a knight stick to 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 focused or occupied by way of one of many queens?

Each major subject is handled intensive from its ancient belief via to its prestige this day. Many attractive options have emerged for simple chessboard difficulties on the grounds that mathematicians first all started engaged on them in earnest over 3 centuries in the past, yet such difficulties, together with these concerning polyominoes, have now been prolonged to three-d chessboards or even chessboards on strange surfaces equivalent to toruses (the an identical of taking part in chess on a doughnut) and cylinders. utilizing the hugely visible language of graph idea, Watkins lightly courses the reader to the leading edge of present study in arithmetic. through fixing a few of the many routines sprinkled all through, the reader can percentage 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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17 A hexagonal cross-section and four mini-tours. Another solution uses an idea given by Ian Stewart in Another Fine Math You’ve Got Me Into [28]. At the heart of this solution is the following very beautiful idea from geometry: if you hold a 2 × 2 × 2 cube so that one of its main diagonals is vertical and then cut the cube exactly in half with a horizontal slice, the cross-section you get will be a hexagon. 17. Now, here is the nice surprise. This hexagon just happens to form a small knight’s mini-tour, which has been labeled A1 , A2 , .

17. 17 A bishop’s ‘tour’. 18. 6 There are two key observations to be made: the first is that if we imagine the king tracing a curve as he moves along his path, then this simple closed curve will separate the plane into two parts, one part inside the curve and one part outside the curve; thus, the king must visit the perimeter squares of the chessboard in order ; otherwise, intermediate perimeter squares would become isolated. 18 A queen’s tour. 19 A king’s tour with fourteen horizontal moves and fourteen vertical moves.

17. Furthermore, since a cube has four main diagonals, we can in this way construct four distinct hexagons and, hence, four distinct knight’s mini-tours. The remaining three mini-tours have been labeled B1 , B2 , . . , B6 ; C1 , C2 , . . , C6 ; and D1 , D2 , . . 17. These minitours are, of course, closed 6-cycles which must be joined together in some way. But, note that A6 –B1 , B6 –C1 , C6 –D1 , and D6 –A1 are all legal knight’s moves on the 2 × 2 × 2 cube. Thus, we can form the following knight’s tour of the entire 2 × 2 × 2 cube: A1 , A2 , .

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