Superpolynomial period lengths of the winning positions in the subtraction game

Abstract

AbstractGiven a finite set of positive integers, A, and starting with a heap of n chips, Alice and Bob alternate turns and on each turn a player chooses $$x\in A$$ x ∈ A with x less than or equal to the current number of chips and subtract x chips from the heap. The game terminates when the current number of chips becomes smaller than $$\min \{A\}$$ min { A } and no moves are possible. The player who makes the last move is the winner. We define $$w^A(n)$$ w A ( n ) to be 1 if Alice has a winning strategy with a starting heap of n chips and 0 if Bob has a winning strategy. By the Pigeonhole Principle, $$w^A(n)$$ w A ( n ) becomes periodic, and it is easy to see that the period length is at most an exponential function of $$\max \{A\}$$ max { A } . The typical period length is a linear function of $$\max \{A\}$$ max { A } , and it is a long time open question if exponential period length is possible. We consider a slight modification of this game by introducing an iitial seed S that tells for the few initial numbers of chips whether the current or the opposite player is the winner, and the game ends when the first such position is achieved. In this paper we show that the initial seed cannot change the period length of $$w^A(n)$$ w A ( n ) if the size of A is 1 or 2, but it can change the period length with $$|A|\ge 3$$ | A | ≥ 3 . Further, we exhibit a class of sets A of size 3 and corresponding initial seeds such that the period length becomes a superpolynomial function of $$\max \{A\}$$ max { A } .