Counting Knight’s Tours through the Randomized Warnsdorff Rule

We give an estimate of the number of geometrically distinct open tours G for a knight on a chessboard. We use a randomization of Warnsdorff rule to implement importance sampling in a backtracking scheme, correcting the observed bias of the original rule, according to the proposed principle that “most solutions follow Warnsdorff rule most of the time”. After some experiments in order to test this principle, and to calibrate a parameter, interpreted as a distance of a general solution from a Warnsdorff solution, we conjecture that G = 1.22 × 10. Instituto de Computación, Facultad de Ingenieŕıa, Universidad de la República. Address: Julio Herrera y Reissig 565, Montevideo, Uruguay. Código Postal: 11300. Casilla de Correo N. 30. e-mail: [email protected] Centro de Matemática, Facultad de Ciencias, Universidad de la República. Address: Facultad de Ciencias, Iguá 4225, Código Postal: 11400, Montevideo, Uruguay. e-mail: [email protected]