## 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.

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THE FOUR KNIGHTS' TOURS.
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THE FOUR KANGAROOS.