Mathematical_Recreations-Kraitchik-2e

~66 Mathematical Recreations
6. Many generalizations of the knight's problem have
been proposed. Many alterations of the size and shape of
the board have already been considered. One may also form
a chessboard of unusual shape by covering part of the cells
on the usual 8 X 8 board.
The move of the knight may also be generalized. Instead
of using the components 2 and
1 for his move, other components
may be taken. Figure
140 shows three tours of the
24 white cells of the chessboard
of order 7, using components
3 and 1. Such a knight cannot
move from one color to another.
The first tour is closed;
the others are symmetric with
FIGURE 142. respect to the center. In
Figure 141 are given two such
tours on an ordinary board, and in Figure 142 a tour on a
board on which no tour can be made by an ordinary knight.
We may also give the knight a double move, using say the
components (8, 1) or (4, 7) at will, since 8 2 + }2 = 42 + 72•
Or we may allow him to move with components (4, 3) or
(5, 0). This last knight can move from al to a6, to fI, to d3,
or to c4. Wherever he may be on the usual chessboard, he
has a choice of just 4 cells.
~uber-Stockar proposed another generalization by permitting
the knight 8 other moves, according to his position.
The components of his move change from place to place. He
is not required to change from one color to the other. With
this knight, closed routes are possible even when the number
of cells is odd.