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 isn't easily approximately chess however the chessboard itself--that uncomplicated grid of squares so universal to video games worldwide. And, extra importantly, the attention-grabbing arithmetic in 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 strangely fertile zone 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 concentrated or occupied via one of many queens?

Each major subject is taken care of extensive from its ancient belief via to its prestige this present day. Many appealing strategies have emerged for easy chessboard difficulties for the reason that mathematicians first 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 third-dimensional chessboards or even chessboards on strange surfaces corresponding to toruses (the identical 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. via fixing the various many workouts sprinkled all through, the reader can proportion absolutely within the pleasure of discovery.

Showing that chess puzzles are the place to begin 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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Extra info for Across the Board: The Mathematics of Chessboard Problems

Example text

D6 , A1 . 18. 18 Covering the 8 × 8 board with tetrominoes. On the other hand, it is impossible to cover the entire 8 × 8 board using only Z tetrominoes. Suppose you were to start by covering the upper left-hand corner square with a Z tetromino placed horizontally. This would then force you to place another one right next to it, also along the top row, and then a third one is forced as well. 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.

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. 2 An incomplete open tour. 3 Closing the tour. the time being the four missing squares a, b, c, and d). 3 that we can form a closed tour of these 60 squares by merely tracing the original tour from 1 to 51, then jumping to 60 and working backwards to 52 and finally jumping back to 1.

So let’s begin this section by seeing why there are no tours for the 3 × 6 or the 3 × 8 boards. The 3 × 4 board, which also has no tour, we will leave for later as a special case of another situation, and so we will ignore it for now. The 3 × 6 Chessboard For the 3 × 6 board we will use an easy idea from graph theory. I’ll describe the idea first in terms of necklaces. 1 Removing two vertices from the knights graph of the 3 × 6 board. if the two links happened to be right next to one another); three links, three pieces; and so on.

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