On the knight\u27s tour on the fifteen puzzle and the topspin puzzle

This thesis is about two types of puzzles. It is based on the two articles: Knight\u27s Tour on the 15-Puzzle by Spencer P. Hurd and David A. Trautman, Permutation Puzzles by John H. Wilson. The first puzzle is called the Fifteen Puzzle . The 15-puzzle is a 4-by-4 array of cells that are labeled from 1 to 15, where the 16th cell is the blank cell. A numbered cell can be moved by sliding each block. This puzzle was then integrated with the knight\u27s tour. Using L-shaped moves like those of the knight on the chessboard, a knight\u27s tour is possible when the sequence of moves lands on all the cells only once and ends at the last cell. A tour is legal if it is possible to transform it into the solution diagram. We determined that there are 320 Knight\u27s tours on the fifteen puzzle of which 160 are legal tours and 160 are illegal tours. The second puzzle is the Topspin puzzle. Here there are N numbered disks packed in an oval track of which T \u3c N disks are fit into the turnstile. We call this the [T, N] puzzle. Three moves are available at any time namely, the shift right, shift left and flip moves. The shift right move will slide all the numbers one position clockwise around the block without changing the arrangement while the shift left moves slides all the numbers one position counterclockwise around the block without changing the arrangement. The flip move turns the turnstile 180 degrees. Each move corresponds to a permutation of the disks. The puzzle is solvable if the 3 moves could generate all the permutations of Sn. By using concepts in abstract algebra, we determined that the [4, 20] puzzle is solvable and so is the [2, N] puzzle for N \u3e 2. We also determined some values for [T, N] that are not solvable