How good is the Warnsdorff's knight's tour heuristic?
  Warnsdorffs rule for a knights tour is a heuristic, i.e., it is a rule that does not produce the desired result all the time. It is a classic example of a greedy method in that it is based on a series of locally optimal choices. This note describes an analysis that determines how good the heuristic is on an 8 X 8 chessboard. The order of appearance in a permutation of the eight possible moves a knight can make determines the path the knight takes. A computer analysis is done of the 8! permutations of the order of a knights moves in Warnsdorffs rule on an 8 X 8 chessboard for tours starting on each of the 64 squares. Whenever a tie occurs for moves to vertices that have the lowest degree, the first of these vertices encountered in the programming loop is chosen. The number of permutations of the 8! total that yield non-Hamiltonian paths is tallied. This will be the same value if we consistently choose the last of these vertices encountered. 
Samuel L. Marateck