board on which a re-entrant tour is possible is one that is 6 by 5.

A complete knight's path (not re-entrant) over all the squares of a
board is never possible if there be only two squares on one side; nor is
it possible on a square board of smaller dimensions than 5 by 5. So that
on a board 4 by 4 we can neither describe a knight's tour nor a complete
knight's path; we must leave one square unvisited. Yet on a board 4 by 3
(containing four squares fewer) a complete path may be described in
sixteen different ways. It may interest the reader to discover all
these. Every path that starts from and ends at different squares is here
counted as a different solution, and even reverse routes are called
different.


339.--THE FOUR KNIGHTS' TOURS.

[Illustration]

I will repeat that if a chessboard be cut into four equal parts, as
indicated by the dark lines in the illustration, it is not possible to
perform a knight's tour, either re-entrant or not, on one of the parts.
The best re-entrant attempt is shown, in which each knight has to
trespass twice on other parts. The puzzle is to cut the board
differently into four parts, each of the same size and shape, so that a
re-entrant knight's tour may be made on each part. Cuts along the dotted
lines will not do, as the four central squares of the board would be
either detached or hanging on by a mere thread.


340.--THE CUBIC KNIGHT'S TOUR.