Mathematical_Recreations-Kraitchik-2e

88 Mathematical Recreations
ing l's and O's in such a way that the new sums of digits are
even in every column. Since the first:digit from the left which
was altered in the number was reduced, this means that the
new number is less than the original, and hence the change
in digits is equivalent to a subtraction. Thus a single subtraction
from just one of the three numbers, or taking a chip
or chips from one pile, alters an unstrategic position to a position
of the strategic type.
To complete the proof of the theorem we have only to observe
that the winning position (0, 0, 0) is of the same special
type as the strategic positions. As a result of the first two
paragraphs of the proof it is clear that once a player captures
a position of this special type he cannot be dislodged from it,
and it is equally clear that any sequence of such positions
must ultimately lead to the winning position. Furthermore,
these special positions are the only ones which can be maintained.
Thus these are the only strategic positions.
It is clear that every sequence of strategic positions must
ultimately lead to a position of the type (0, n, n). From here
the sequence must lead to (0, 1, 1) and then to (0, 0, 0).
Here is a list of the strategic positions [other than those
of type (0, n, n)] for any game in which each pile has at most
15 chips.
1, 2, 3 2, 4, 6 3, 4, 7 4, 8,12 5, 8,13 6, 8,14 7, 8,15
1, 4, 5 2, 5, 7 3, 5, 6 4, 9,13 5, 9,12 6, 9,15 7, 9,14
1, 6, 7 2, 8,10 3, 8,11 4,10,14 5,10,15 6,10,12 7,10,13
1, 8, 9 2, 9,11 3, 9,10 4,11,15 5,11,14 6,11,13 7,11,12
1,10,11 2,12,14 3,12,15
1,12,13 2,13,15 3,13,14
1,14,15
In general the initial position is not strategic, so the player
who begins has the advantage, since he can make his position
strategic. Only if the initial position is strategic is the advantage
with the second player.