Mathematical_Recreations-Kraitchik-2e

CHAPTER ELEVEN
THE PROBLEM OF THE KNIGHT
THE strange move of the knight in chess makes his operations
particularly fascinating. He is allowed to occupy any unoccupied
space on the board which is two columns and one
row or two rows and one column away from the cell he is in,
regardless of whether or not the intervening cells are occupied.
Almost the first question one would be likely to ask on
learning of this irregular move is whether he can reach every
cell on the board. When this has been answered in the affirmative,
one might well ask what strange sort of journey he
would make in going just once to every cell, if that is possible.
It is possible, and the investigation of these "grand tours"
is the subject of this chapter. (If the reader finds any of our
notations or terminology unfamiliar, he is advised to glance
through Chapter Ten on the Problem of the Queens.)
1. A solution is called re-entrant or closed if the knight can
be brought back to his initial position by one more move;
otherwise the solution is open.
A solution is usually described by placing in each cell the
number indicating the order in which that cell was reached,
the initial cell being denoted by 1. Since the color of the cell
on which the knight lands changes at each move, all oddnumbered
cells are of one color, all even-numbered cells of
the other.
It follows that if a board of a particular shape has one
more cell of one color than of the other, a solution (if it exists)
must be open. Hence all solutions on boards of odd order
are open. But if the difference in the numbers of cells of the
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