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The Knight's Tour is a mathematical problem involving a knight on a chessboard. The knight is placed on the empty board and, moving according to the rules of chess, must visit each square exactly once.
There are several billion solutions to the problem, of which about 122,000,000 have the knight finishing on a square from which it attacks the starting square. Such a tour is described as closed. Otherwise the tour is open (as in the diagram).
Many variations on this topic have been studied by mathematicians, including Euler, during the centuries using:
- differently sized boards
- two-player games based on this idea
- problems using slight variations on the way the knight moves.
The knight's tour problem is an instance of the more general Hamiltonian path problem in graph theory, which is NP-complete. The problem of getting a closed knight's tour is similarly an instance of the hamiltonian cycle problem. Note however that, unlike the general Hamiltonian path problem, the knight's tour problem can be solved in linear time (Conrad et al 1994).
The pattern made by a Knight's Tour has often been used as a literary constraint. The earliest instance of this is found in Rudrata's Kavyalankara written during the 9th century.
In the 20th century the Oulipo group of writers used it among many others. The most notable example is the 10 × 10 Knight's Tour which sets the order of the chapters in Georges Perec's novel A User's Manual.
Schwenk's Theorem
For any m × n board with m less than or equal to n, a closed knight's tour is possible unless one or more of these three conditions are true:
1. m and n are both odd
2. m = 1, 2, or 4
3. m = 3 and n = 4, 6, or 8
It is not hard to show that when condition 1 is true a closed knight's tour is impossible.
Imagine a chessboard colored black and white in the manner that we are all familiar with (alternating). Whenever a knight moves it lands on the opposite color. If it is on white, it moves to black. If it is on black, it moves to white.
Since m and n are both odd, the number of white squares and black squares are different. It follows that a closed tour is impossible on such a board. Open tours may still exist.
It is also easy to see why when m = 1 or 2 that no knight's tour is possible: the knight would not be able to reach every square (with the exception of the trivial case 1x1).
See also
References
- A. Conrad, T. Hindrichs, H. Morsy, and I. Wegener. "Solution of the Knight's Hamiltonian Path Problem on Chessboards." Discrete Applied Math, volume 50, no.2, pp.125-134. 1994.
- Watkins, John J. Across the Board: the Mathematics of Chessboard Problems. Princeton UP, 2004.
External links
Chess problems | Graph theory | Recreational mathematics
Springerproblem | Problema del caballo | Problème du cavalier | ナイト・ツアー | Skakačev obhod | 騎士巡邏
"Knight's tour"