THE BOARD IN COMPARTMENTS.
(
The Guarded Chessboard)
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
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.
Answer:
In attempting to solve this problem it is first necessary to take the
two distinctive compartments of twenty and twelve squares respectively
and analyse them with a view to determining where the necessary points
of entry and exit lie. In the case of the larger compartment it will be
found that to complete a tour of it we must begin and end on two of the
outside squares on the long sides. But though you may start at any one
of these ten squares, you are restricted as to those at which you can
end, or (which is the same thing) you may end at whichever of these you
like, provided you begin your tour at certain particular squares. In the
case of the smaller compartment you are compelled to begin and end at
one of the six squares lying at the two narrow ends of the compartments,
but similar restrictions apply as in the other instance. A very little
thought will show that in the case of the two small compartments you
must begin and finish at the ends that lie together, and it then
follows that the tours in the larger compartments must also start and
end on the contiguous sides.
In the diagram given of one of the possible solutions it will be seen
that there are eight places at which we may start this particular tour;
but there is only one route in each case, because we must complete the
compartment in which we find ourself before passing into another. In any
solution we shall find that the squares distinguished by stars must be
entering or exit points, but the law of reversals leaves us the option
of making the other connections either at the diamonds or at the
circles. In the solution worked out the diamonds are used, but other
variations occur in which the circle squares are employed instead. I
think these remarks explain all the essential points in the puzzle,
which is distinctly instructive and interesting.
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