Mathematical_Recreations-Kraitchik-2e

86 Mathematical Recreations
innings or who gets first choice in choosing teams for a scratch
game of baseball. But neither contestant knows the play
limits of his opponent, nor is the first position fixed by these
limits, so the result is pretty much a matter of chance. There
is also a European custom of a similar nature, called in Belgium
by the Walloon word" guiser." In such games as soccer
or hockey, in order to determine which team defends
which goal or which team first takes the offensive, the two
captains take positions a little apart. Then they take turns
in advancing toward each other, each by the length or width
of his own foot. The first to place his foot on that of his adversary
wins.
If the battle of numbers be played with more than two
players it loses its mathematical interest, for the result of the
game cannot be decided by the first moves. But it must be
admitted that in this case the loss of mathematical interest
is more than compensated by the gain in sporting interest,
for who wants to play for long a game whose result he can
always foresee?
2. NIM. This game may be regarded as a generalization
of the battle of numbers, and, like it, has a complete mathematical
analysis.
The game is played by two persons, playing alternately.
An arbitrary number of objects, say chips, is distributed into
just three heaps in an arbitrary way. Each player in his
turn is to take as many chips as he wishes (but at least one)
from any heap he wishes. The last one to remove a chip is
the winner.
We define the expressions position, holding a position, and
strategic position as in the battle of numbers. A position will
be denoted by the set of three numbers (a, b, c) representing
in order the number of chips in each pile. In nim, as in the
battle of numbers, strategic positions are independent of the
player holding them, and are characterized by the fact that