Mathematical_Recreations-Kraitchik-2e

Games 269
The checker man moves one step at a time along a diagonal,
but always in such a way that he changes rows in the
same direction. Hence we need only be concerned with what
column he occupies. Let us suppose that he is a White piece
and starts at the cell (5, 1), moving up the board. We may
mark this cell with a 1 to indicate that there is only one way
for him to get there. Similarly we mark with a 1 the two
cells that he can reach on his first move, (4, 2) and (6, 2).
1 7 21 35 35 21 7 1
1 6 15 20 15 6 1
1 5 10 10 5 1
1 4 6 4 1
1 3 3 1
1 2 1
1 1
1
FIGURE 144. Pascal's Triangle in Checker Moves.
At his second move he can reach the cell (3, 3) only from
(4, 2); he can reach (5, 3) either from (4, 2) or (6, 2); and
he can reach (7, 3) only from (6, 2). We mark these cells 1,
2, 1, respectively. At his third move he can reach the cells
(2, 4) and (8, 4) in only one way. But he can reach (4, 4)
either from (3, 5) or from (3, 3). Since there are 2 ways of
reaching (3, 5) and 1 of reaching (3, 3), this gives him 3 ways
of reaching (4, 4) from (5, 1). Similarly for (4, 6). This is
sufficient to show how we may proceed from here on. Suppose
we have marked every cell that he can reach in the kth
row. Then the number of ways in which he can reach a cell
in the next row is always the sum of the numbers in the cells
in the kth row from which he can move to the new cell. In
this way we get an array of numbers such as that in Figure
144, in which every number is the sum of the two nearest
numbers in the row below.