Question 34
Let $N \ge 4$ be a fixed integer. Two players $A$ and $B$ write down numbers, each number in continuation to the previous expression. First $A$ writes $+1$ or $-1$, then $B$ writes $+2$ or $-2$, then $A$ writes $+3$ or $-3$ etc.; at step $k$ the player to move must write $+k$ or $-k$. The objective of each one is that after a move of his several consecutive numbers in the obtained expression, taken with their signs, have sum divisible by $N$. For each $N$ determine which of the players has a winning strategy, if any.