Question 75
Let $m$ be a positive integer, $n = 2^m - 1$, and $P_n = \{1, 2, \dots, n\}$ be the set of $n$ points on the number axis. A grasshopper jumps between adjacent points on $P_n$. Find the maximal number of $m$ such that for any $x, y \in P_n$, the number of ways that a grasshopper jumping from $x$ to $y$ by 2012 steps is even (passing $x$ or $y$ on the way is permitted). (posed by Zhang Sihui)