Problem

Source:

Tags: combinatorics, game, romania, Romanian TST, TST



Let $N\geq 4$ be a fixed positive integer. Two players, $A$ and $B$ are forming an ordered set $\{x_1,x_2,...\},$ adding elements alternatively. $A$ 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 the $k^{th}$ step, the chosen number must always be $k$ or $-k$) The winner is the first player to make the sequence sum up to a multiple of $N.$ Depending on $N,$ find out, with proof, which player has a winning strategy.