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 basic grid of squares so universal to video games all over the world. And, extra importantly, the interesting arithmetic at the back of it. From the Knight's travel challenge and Queens Domination to their many adaptations, John Watkins surveys the entire famous 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 way of one of many queens?

Each major subject is handled extensive from its ancient perception via to its prestige this present day. Many appealing recommendations have emerged for uncomplicated chessboard difficulties considering that mathematicians first all started engaged on them in earnest over 3 centuries in the past, yet such difficulties, together with these regarding polyominoes, have now been prolonged to 3-dimensional chessboards or even chessboards on strange surfaces similar to toruses (the similar of enjoying chess on a doughnut) and cylinders. utilizing the hugely visible language of graph idea, Watkins lightly publications the reader to the vanguard of present examine in arithmetic. by way of fixing a few 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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Additional resources for Across the Board: The Mathematics of Chessboard Problems

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But then you are really completely stuck, since there is no way to cover the remaining two squares at the end of the top row. Of course, it makes no difference if you begin in the upper left-hand corner by orienting the first tetromino vertically. The same difficulty arises along the left edge. It is also impossible to cover either a 4 × 5 or a 2 × 10 board with the five distinct polyominoes. A checkerboard coloring of either of these boards has ten white squares and ten black squares. Now, it is easy to see that no matter how you place them on the board, the square tetromino, the straight tetromino, the L tetromino, and the Z tetromino each cover exactly two white squares and two black squares.

By being careful to never inadvertently isolate a vertex in this way, it is relatively easy to conclude that there are only two ways to complete this cycle, namely, BCPNMDFKLTSRQZXWVJHGB and BCPNMDFGHXWVJKLTSRQZB. 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.

The bold edges then connect these individual open tours into a tour of the entire board. In fact, you can even think of this process as having started in the center with a knight’s tour of the 1×1 chessboard! 10 Open knight’s tours for the (4n + 1) × (4n + 1) chessboards. 11 An attempt at an open tour of the 4 × 4 chessboard. 4 No, not even an open tour is possible. 11. Earlier we argued that the 4 × 4 board doesn’t have a closed knight’s tour because vertices 1 and 3 each have degree 2, which would force the four edges of a 4-cycle, namely, edges 1–2, 2–3, 3–4, and 4–1, to be in any Hamiltonian cycle of the entire graph, which is clearly impossible.

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