Mathematical_Recreations-Kraitchik-2e

The Problem of the Knight
U9
2. W ARNSDORF'S RULE. Among the many methods for
obtaining knight's tours, perhaps the best is by Warnsdorf's
rule: At every move place the knight in the cell from which
there are the fewest exits to unoccupied cells.
In applying Warnsdorf's rule we find many cases where
the knight has a choice of two cells, each having the same
minimum number of exits. Each choice offers two alternative
solutions. However, the total number of solutions obtainable
by this method is not very great, and we may even
be able to find all the solutions corresponding to a given initial
cell. Warnsdorf's method gives only a few particular
solutions.
Although it is laborious to compute every time the number
of possible exits, this inconvenience is counterbalanced by
the advantage of a surprising property that is very difficult
to analyze, namely: the many mistakes that it is difficult to
avoid do not prevent one from finishing the tour, except in
certain cases.
The rule is a rule of common sense, and is applicable to all
chessboards. It is also good if the move of the knight is
changed.
3. In this and the succeeding paragraph we consider possible
routes on small rectangular boards. Here we assume
that neither dimension is 4.
If one dimension is < 3 no tour is possible.
If one dimension is 3 the other must be ~ 7. If the second
dimension is even and ~ 10, closed solutions exist.
If one dimension is 5 the other must be as great; and if the
second dimension is even, there are closed tours.
If one dimension is 6 the other must be ~ 5. Closed tours
will then exist.
If one dimension is 7 or more, a tour can always be found.
Closed routes are possible in general when the total number
of cells is even and neither dimension is 4.