Which Boards Have Tours
Schwenk proved that for any m × n board with m less than or equal to n, a closed knight's tour is always possible unless one or more of these three conditions are met:
- m and n are both odd; n is not 1
- m = 1, 2, or 4; n is not 1
- m = 3 and n = 4, 6, or 8.
Cull and de Curtins proved that on any rectangular board whose smaller dimension is at least 5, there is a (possibly open) knight's tour.
Read more about this topic: Knight's Tour
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