338.--THE BOARD IN COMPARTMENTS.

[Illustration]

We cannot divide the ordinary chessboard into four equal square
compartments, and describe a complete tour, or even path, in each
compartment. But we may divide it into four compartments, as in the
illustration, two containing each twenty squares, and the other two each
twelve squares, and so obtain an interesting puzzle. You are asked to
describe a complete re-entrant tour on this board, starting where you
like, but visiting every square in each successive compartment before
passing into another one, and making the final leap back to the square
from which the knight set out. It is not difficult, but will be found
very entertaining and not uninstructive.

Whether a re-entrant "tour" or a complete knight's "path" is possible or
not on a rectangular board of given dimensions depends not only on its
dimensions, but also on its shape. A tour is obviously not possible on a
board containing an odd number of cells, such as 5 by 5 or 7 by 7, for
this reason: Every successive leap of the knight must be from a white
square to a black and a black to a white alternately. But if there be an
odd number of cells or squares there must be one more square of one
colour than of the other, therefore the path must begin from a square of
the colour that is in excess, and end on a similar colour, and as a
knight's move from one colour to a similar colour is impossible the
path cannot be re-entrant. But a perfect tour may be made on a
rectangular board of any dimensions provided the number of squares be
even, and that the number of squares on one side be not less than 6 and
on the other not less than 5. In other words, the smallest rectangular