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magic tour
Suppose a chess piece makes a tour on an n × n chessboard whose squares are numbered from 1 to n2 along the path of the piece. The tour is a magic tour if the resulting arrangement of numbers is a magic square, and is a semimagic tour if the resulting arrangement of numbers is a semi-magic square. Magic knight's tours aren't possible on n × n boards if n is odd. They are possible for all boards of size 4k × 4k for k > 2, but are believed to be impossible for n = 8.
Magic tours have been found in 4 × 4 × 4, 8 × 8 × 8, and 12 × 12 × 12 cubes, and on the surface of 8 × 8 × 8 cube. However, there are no known knight tours in hypercubes.
[Thanks to Awani Kumar for corrections to and additional information for this entry.]
Reference
- Kumar, Awani. "In Search of Perfect Magic Tours of Knight on 12×12 Board." The On-line Journal for Mathematical Recreations, Issue 26, April-June 2003.
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• MATHEMATICS
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