Mathematical_Recreations-Kraitchik-2e

274 Mathematical Recreations
better to use a modification of the marking procedure. This
time we shall have to mark each cell with two numbersthe
first, which we shall call the order of the cell, to indicate
the minimum number of moves required to reach it, and the
second, the multiplicity of the cell, to tell the number of
shortest routes to it. To do this, first mark the initial cell
with the order 0, then mark the cells accessible from it in one
FIGURE 147.
Algebra of the King's llfove, in Chess: 8 X 8 Board.
move with the order 1, then all cells (not already marked)
accessible from cells of order 1 in one move with the order 2,
and so on. Then we assign multiplicity 1 to the initial cell
and to all the cells of order 1. The multiplicity of a cell of
order 2 is then the sum of the multiplicities of the cells of
order 1 from which it can be reached in one move. Mter the
multiplicities of the cells of order 2 have been determined we
can find the multiplicity of a cell of order 3 by adding the
multiplicities of all cells of order 2 from which it is accessible,
and so on. Figures 148 and 149 give the results for the first
few moves on a limited board, and Figure 150 the results
for the "\Vhite knight's moves on the ordinary board starting
from his own cell.
How many different chess games can there be?
\Vhite cun begin with anyone of 20 possible first moves