502 Combinatorial games
17.1 Subtraction games ..................
Starting with a given positive integerN, two players alternately subtract
a positive amount less than a given positive numberd<N. The one
who gets down to 0 wins.
Theorem 17.1.The player who secures a multiple ofdhas a winning
strategy.
An equivalent version: the battle of numbers
Starting with 0, two players alternately add positive integers less than a
given limitd. The one who gets to a specifiedNwins.
The winning positions are the terms of the arithmetic progression of
common differencedcontainingN. Specifically, the small numbermod
(N, d)is a winning position. Therefore, the first player has a winning
strategy if and only ifNis not divisible byd.
