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 all over the world. And, extra importantly, the interesting arithmetic in 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 unusually fertile sector of leisure arithmetic. Can a knight persist with a direction 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 means of one of many queens?

Each major subject is handled intensive from its ancient perception via to its prestige this present day. Many attractive options have emerged for easy chessboard difficulties for the reason that mathematicians first begun 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 third-dimensional chessboards or even chessboards on strange surfaces akin to toruses (the an identical of enjoying 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. via fixing a number of the many workouts sprinkled all through, the reader can proportion absolutely within the pleasure of discovery.

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

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In other words, the knight has to avoid squares colored B and D entirely in order to visit all of the squares colored A. Therefore, no knight’s tour is possible. Incidentally, this color pattern was also used somewhat earlier by Maurice Kraitchik to show that a knight’s tour is impossible for the 4 × n board, but his argument was considerably more involved [20]. Schwenk’s Theorem At this point we really have seen all of the chessboards that don’t have a knight’s tour. All other rectangular boards have tours!

13 shows the ‘board’ for the Icosian Game, that is, a graph of the dodecahedron. Complete a Hamiltonian cycle that begins with the five vertices B, C, P, N, and M. Chessboards can certainly be toured by chess pieces other than the knight. For example, in Chapter 1 we made use of a rook’s tour to prove Gomory’s Theorem. Still, for obvious reasons, the knight is the only chess piece for which mathematical questions about touring have generated much general interest. Not surprisingly, however, there are numerous recreational puzzles involving tours by other pieces.

It must then go to vertex 3, followed next by 4; otherwise vertex 3 will become isolated. The situation concerning where the Hamiltonian path ends is also symmetric and, for the same reason, we may as well assume that it ends at vertex 16, which means that the end of the Hamiltonian path can be assumed to consist of 13–14–15–16. 11, where it is now clear that whichever way the knight decides to go from vertex 4, either left or right, several vertices will inevitably be missed on the way to vertex 16.

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