Mathematical_Recreations-Kraitchik-2e

The Problem of the Knight 263
tal median) to the route from c4 to c3, which begins on c2
and ends on cl. Joining this with our first route we have the
complete tour.
It follows from this that every tour begins and ends on an
1 3 5 9 15 1 5 9
16 14 12 6 2 16 12 6
2 4 8 10 14 4 8 10
15 13 11 7 3 13 11 7
14 4 6 10 1 30 3 20 5 22 9 26
3 13 9 7 16 19 14 29 12 25 6 23
15 1 5 11 31 2 17 4 21 8 27 10
2 16 12 8 18 15 32 13 28 11 24 7
FIGURE 137.
outside cell, so no tour on such a chessboard can be closed.
Because of its special properties it is easy to count the
FIGURE 138.
number N of all possible tours on a board of dimension 4k.
Thus we have:
k 3 4 5 6 7 8 9 10
N 8 ° 82 744 6,378 31,088 189,688 1,213,112.