Mathematical_Recreations-Kraitchik-2e

84 Mathematical Recreations
N ow if strategic position S guarantees victory when S =
Q - (p + 1), then it becomes the object of a player to get in
position S, hence to get into a still earlier position S'. from
which he cannot be prevented from achieving S. Just as the
strategic position S was calculated by subtracting (p + 1)
from Q, so the earlier strategic position S' may be calculated
by subtracting (p + 1) from S; or we may say
S' = Q - 2(p + 1). A still earlier strategic position S" is
equal to Q - 3(p + 1). Reasoning thus backwards from Q
toward the starting position 0, it can be seen that a player can
put himself in a position early in the game from which
(always assuming that he plays correctly) he can proceed to
successive strategic positions and victory despite anything his
opponent may do.
H p = 10 and Q = 100, the strategic positions are 89, 78,
67, 56, 45, 34, 23, 12 and 1. Hence the first to play may make
certain of winning by taking the strategic position 1. But
if he selects any other number, his opponent can call the tune
by occupying the position 12. It is easy to see how one may
make sure of continuing to occupy strategic positions once
he has gained one.
The game seems to have been played originally with the
limits 10 and 100. Bachet de Meziriac (1613) noted that it
could be varied by changing either or both of the limits. He
also remarked that if the game limit is 100, the first player can
always win under the play limit 8, since the first strategic
position is still 1, but with the play limit 9, the second player
can always win since the first strategic position is then 10.
He goes on to recommend playing this game against those
who do not understand its theory j even then, he warns, if
an adversary is clever, one must not always take the strategic
positions, since the adversary will then discover them himself.
Instead, Bachet advises waiting until near the end of
the game before capturing a strategic position.