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 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 unusually fertile zone of leisure arithmetic. Can a knight stick to 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 extensive from its ancient perception via to its prestige this day. Many appealing ideas have emerged for simple chessboard difficulties considering 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 resembling toruses (the similar of taking part in chess on a doughnut) and cylinders. utilizing the hugely visible language of graph conception, Watkins lightly publications the reader to the vanguard of present learn in arithmetic. by means of fixing many of the many routines sprinkled all through, the reader can percentage absolutely within the pleasure of discovery.
Showing that chess puzzles are the place to begin for vital mathematical principles 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 info for Across the Board: The Mathematics of Chessboard Problems
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.
Let’s see how the idea works in a speciﬁc case by showing that any 3 × n board has a knight’s tour for n 10 and n even, that is, all of 45 CHAPTER 3 the boards, 3×10, 3×12, 3×14, 3×16, 3×18, . . —this sequence goes on forever—have tours. 5 with tours for the 3×10 and the 3×12 boards. Normally, in an inductive argument like this, we would just start with the smallest board, namely, the 3 × 10 board, but we actually need to begin with tours of both of these boards in this case, as you will see. 5 Knight’s tours for the 3 × 10 and the 3 × 12 boards.
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. We then, of course, unfold the graph. 14 Switch the knights in sixteen moves. We can immediately see that it takes seven moves just to move the three black knights into the correct position; and, by symmetry, there is really only one way to do this: bring the knight at 5 straight down to 8 and move the other two knights two squares each, that is from 3 to 1 and from 12 to 10.
Across the Board: The Mathematics of Chessboard Problems by John J. Watkins