Symmetryplus 68

6 SYMmetryplus 68 Spring 2019 This square is a white square so it cannot be the starting or ending square for the tour. In addition, the knight can only go to or come out of this square via two squares, the middle square in the top row and the middle square on the bottom. Without loss of generality, I will label the square in the top row as the position the knight was in prior to x and the square in the bottom row as the one it goes to immediately after x . (Of course, it could have been the other way around but the alternative is a mirror image of the one being considered, and so anything I say about this one, will be true of the alternative!) x-1 A x x+1 However, when will square A be visited? A is only linked by knight moves to the squares labelled as the x – 1 th and x + 1 th positions. It needs to be somewhere in the tour, and this means that A must either be positioned at the x – 2 th or x + 2 th position of the tour. We are now stuck! If A is at the x – 2 th position there are no positions left for it to have been somewhere previous to this and so A must be the 1 st position. If A is at the x + 2 th position it cannot go anyway else after this and so A must be the 15 th position. However, both of these scenarios mean that the knight’s tour must either start or finish on a white square which we know is impossible. Hence a knight’s tour for a 3 by 5 grid is not possible. However, on a more encouraging note, the 3 by 7 , 3 by 8 and 3 by 9 cases are possible as show below. 17 14 19 2 5 8 11 20 1 16 13 10 3 6 15 18 21 4 7 12 9 5 2 19 22 9 12 17 14 20 23 6 3 18 15 8 11 1 4 21 24 7 10 13 16 5 2 17 14 11 8 19 22 25 16 13 6 3 18 27 24 9 20 1 4 15 12 7 10 21 26 23 The questions I leave the reader to decide are whether the 3 by 6 board has a knight’s tour, and also whether all longer boards of width 3 have knight’s tours. These cases will be followed up in a forthcoming article! Andrew Palfreyman THE KING WHO LIKED CHESS There is a story that a king so liked the game of chess, he decided to give the person who invented the game a reward. The king asked that person what they would like. The inventor, being a canny mathematician asked for some wheat: to be modest they asked for just one grain for the first square of the board, two grains for the second, four grains for the third and so on, doubling each time until the inventor had a reward for each square. The king thought this an excellent modest idea until the total amount was calculated as 18, 446,744,073,709,551,615 grains of wheat! (~1645 times the global wheat production in 2014)