Mathematical_Recreations-Kraitchik-2e

~58 Mathematical Recreations
two colors is greater than 1, then no solution is possible.
(For example, it is not possible to visit all the cells of the
board in Figure 132, which contains 32 black and 25 white
cells.) If the squares are not colored it is often more difficult
to tell whether a tour is possible.
A solution will be called symmetric
to the center if the sum
or difference of the numbers of
each pair of cells symmetric
to the center is constant, using
the sum or dtlIerence according
as the order of the board is
odd or even.
A solution is symmetric to
a median (central horizontal
FIGURE 132.
or vertical line) if the difference
of the numbers of each pair
of cells symmetric with respect to the median is constant.
Two cells symmetric with respect to a median are of different
colors, so the difference of their numbers must be odd, say
2k + 1. Hence the total number of cells is 2(2k + 1), which
shows that such solutions are not possible on ordinary chessboards,
but only on those having 4k + 2 cells and having a
median of symmetry.
Solutions may also be represented geometrically by drawing
on the surface of the board a broken line joining the
centers of the successive squares visited by the knight. A solution
represented in this manner is called doubly symmetric
if it is not changed by a rotation through one or more right
angles. Such a solution is not possible on the ordinary chessboard,
but may be realized on certain boards having 4k cells
and a center of symmetry.
A symmetric solution represented in this geometric manner
is not changed by a rotation through two right angles.