Mathematical_Recreations-Kraitchik-2e

Games 271
left," namely" no change." If we collect terms our expression
becomes u-2 + 2 + u 2 , which means: "Either he moves
2 to the left in just one way or he gets to the original column
in either of 2 ways or he moves 2 to the right." Similarly,
the result of 3 moves may be expressed by (U-l + U)3 =
u-3 + 3u-1 + 3u + u 3, which tells us that he either gets 3 columns
to the left (in just one way) or 1 column to the left (in
any of 3 ways) or 1 column to the right (in any of 3 ways) or
,.
FIGURE 145. Algebra of the Checkerboard.
3 columns to the right (in just one way); and any positive
power can be interpreted similarly.
If now we wish to take into account the fact that when the
man reaches the first or the 8th column he has only one alternative,
if he is to stay on the board, we can do so in either
procedure. In marking the cells we simply do not mark any
cells outside the limits of the board, but continue to mark
the cells on the board according to the same rules, using only
the cells already marked. If we use the algebraic method
we simply drop off as they are formed all terms corresponding
to impossible moves. For example the third move
gives the term u 3 that takes the man to column 8. No
higher power of u may be kept. In forming the fourth
power of (u-1 + u) we therefore omit the last term and write