On the knight's tour on the fifteen puzzle and the topspin puzzle

Date of Publication

1998

Document Type

Bachelor's Thesis

Degree Name

Bachelor of Science in Mathematics

College

College of Science

Department/Unit

Mathematics and Statistics

Abstract/Summary

This thesis is about two types of puzzles. It is based on the two articles: Knight's 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's tour. Using L-shaped moves like those of the knight on the chessboard, a knight's 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's 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 < 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 > 2. We also determined some values for [T, N] that are not solvable.

Abstract Format

html

Language

English

Format

Print

Accession Number

TU08773

Shelf Location

Archives, The Learning Commons, 12F, Henry Sy Sr. Hall

Physical Description

54 leaves

Keywords

Puzzles; Mathematical recreations; Permutations; Algebra, Abstract; Games; Fifteen puzzle

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