Question 53
Fix an integer $N\geq4$. Two players, $A$ and $B$, form a sequence $x_1,x_2,x_3,\ldots$ by adding terms alternately. Player $A$ chooses $x_1$ to be $1$ or $-1$; then $B$ chooses $x_2$ to be $2$ or $-2$; then $A$ chooses $x_3$ to be $3$ or $-3$; and so on. At step $k$, the appended term is $k$ or $-k$, for every $k\geq1$.
The winner is the first player to create a consecutive block of terms whose sum is divisible by $N$; the block may consist of a single term. For each $N$, determine which player has a winning strategy.
The winner is the first player to create a consecutive block of terms whose sum is divisible by $N$; the block may consist of a single term. For each $N$, determine which player has a winning strategy.