Post by Prai JeiPost by George CoxDiscussed here http://mathworld.wolfram.com/MagicTour.html a good site.
Couldn't get in! Access from Freeserve has been blocked because somebody has
been bu--ering about causing multiple hits or something. Help!
Here's the text at least -- I won't attempt to reproduce, or even
describe, the figures!
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Let a chess piece make a tour on an n*n chessboard whose squares are
numbered from 1 to n^2 along the path of the chess piece. Then the
tour is called a magic tour if the resulting arrangement of numbers
is a magic square, and a semimagic tour if the resulting arrangement
of numbers is a semimagic square. If the first and last squares
traversed are connected by a move, the tour is said to be closed (or
"re-entrant"); otherwise it is open. (Note some care with terminology
is necessary. For example, Jelliss terms a semimagic tour a "magic
tour" and a magic tour a "diagonally magic tour.")
[3 diagrams]
Magic knight's tours are not possible on n*n boards for n odd. They
are possible for all boards of size 4k*4k for k > 3, but are believed
to be impossible for n = 8. Beverley (1848) composed the 8*8
semimagic knight's tour (left figure). Another semimagic tour for n =
8 with main diagonal sums of 348 and 168 was found by de Jaenisch
(1862; Ball and Coxeter 1987, p. 185; center figure). The "most
magic" knight's tour known on the 8*8 board has main diagonal sums of
264 and 256 and is shown on the right (Francony 1882). Extensive
histories of knight's magic tours are given by Murray (1951) and Jelliss.
[diagram]
Combining two half-knights' tours one above the other as in the above
right figure does, however, give a magic square (Ball and Coxeter
1987, p. 185).
[diagram]
The illustration above shows a closed magic knight's tour on a 16*16
board (Madachy 1979, p. 88).
[diagram]
A magic tour for king moves is illustrated above (Coxeter 1987, p. 186).
See also:
Chessboard <http://mathworld.wolfram.com/Chessboard.html>,
Knight's Tour <http://mathworld.wolfram.com/KnightsTour.html>,
Magic Square <http://mathworld.wolfram.com/MagicSquare.html>,
Semimagic Square <http://mathworld.wolfram.com/SemimagicSquare.html>,
Tour <http://mathworld.wolfram.com/Tour.html>
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References:
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays
<http://www.amazon.com/exec/obidos/ASIN/0486253570/weisstein-20>,
13th ed. New York: Dover, pp. 185-187, 1987.
Beverly, W. Philos. Mag. p. 102, Apr. 1848.
de Jaenisch, C. F. Chess Monthly. 1859.
de Jaenisch, C. F. Traite des Applications de l'Analyse Mathematiques
au jeu des Echecs. Leningrad, 1862.
Francony. In Le Siécle 1876-1885. (Ed. M. A. Feisthamel). 1882.
Jelliss, G. "Knight's Tour Notes."
<http://home.freeuk.net/ktn/>.
Jelliss, G. "General Theory of Magic Knight's Tours."
<http://home.freeuk.net/ktn/mg.htm>.
Kraitchik, M. l'Echiquier. 1926.
Madachy, J. S. Madachy's Mathematical Recreations
<http://www.amazon.com/exec/obidos/ASIN/0486237621/weisstein-20>. New
York: Dover, pp. 87-89, 1979.
Marlow, T. W. The Problemist. Jan. 1988.
Murray, H. J. R. The Magic Knight's Tours, a Mathematical Recreation. 1951.
Roberts, T. S. The Games and Problems J. Jan. 2003.
Wenzelides, C. Schachzeitung, p. 247, 1849.
Author: Eric W. Weisstein
© 1999 CRC Press LLC, © 1999-2003 Wolfram Research, Inc.
<http://www.wolfram.com/>
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I hope this constitutes "fair use"!
--Odysseus