A sequence of moves by which a
knight visits every
square of a
chessboard exactly once. If the final square is one knight's move away from the initial square of the tour, it is referred to as
re-entrant.
A tour may have several other interesting properties - for example:
- A symmetric tour is one where the path followed by the knight displays two- or four-way symmetry.
- A magic tour is one where numbering the squares from 1 to 64 in the order visited produces a magic square.
The tour may also be on a
board which is not the
standard eight-by-eight size; tours on other boards were studied by
Euler, among others.
An example of a non-reentrant tour, numbering the squares in visiting order:
22 19 44 37 50 35 46 7
43 40 23 20 45 6 49 34
18 21 38 41 36 51 8 47
39 42 17 24 5 48 33 52
16 3 62 57 32 53 28 9
61 58 15 4 25 12 31 54
2 63 60 13 56 29 10 27
59 14 1 64 11 26 55 30
This tour is
semimagic, since all
rows and columns in the array above sum to 260. A "fully" magic square would also have both diagonals summing to 260. In 2003, it was proven through exhaustive computer analysis that a fully magic tour is impossible on the 8x8 board, while 140 semimagic tours are possible.