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Magic Tour

Let a chess piece make a Tour on an $n\times n$ 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. 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. The Magic Constant for the $8\times 8$ Chessboard is 260.

\begin{figure}\begin{center}\BoxedEPSF{MagicToursAlmostKnights.epsf scaled 830}\end{center}\end{figure}

Magic Knight's Tours are not possible on $n\times n$ boards for $n$ Odd, and are believed to be impossible for $n=8$. The ``most magic'' knight tour known on the $8\times 8$ board is the Semimagic Square illustrated in the above left figure (Ball and Coxeter 1987, p. 185) having main diagonal sums of 348 and 168. 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).

\begin{figure}\begin{center}\BoxedEPSF{MagicTour.epsf scaled 1000}\end{center}\end{figure}

The above illustration shows a $16\times 16$ closed magic Knight's Tour (Madachy 1979).


A magic tour for king moves is illustrated above (Coxeter 1987, p. 186).

See also Chessboard, Knight's Tour, Magic Square, Semimagic Square, Tour


Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 185-187, 1987.

Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 87-89, 1979.

© 1996-9 Eric W. Weisstein