Question 4
Let $n$ be a fixed positive odd integer. Take $m+2$ distinct points $P_0,P_1,\ldots,P_{m+1}$, where $m$ is a nonnegative integer, in the coordinate plane satisfying:
1. $P_0=(0,1)$, $P_{m+1}=(n+1,n)$, and for $1\le i\le m$ both coordinates of $P_i$ are integers between 1 and $n$, inclusive.
2. For $0\le i\le m$, segment $P_iP_{i+1}$ is parallel to the $x$-axis if $i$ is even, and to the $y$-axis if $i$ is odd.
3. For every $0\le i\lt j\le m$, the segments $P_iP_{i+1}$ and $P_jP_{j+1}$ share at most one point.
Determine the maximum possible value of $m$.
1. $P_0=(0,1)$, $P_{m+1}=(n+1,n)$, and for $1\le i\le m$ both coordinates of $P_i$ are integers between 1 and $n$, inclusive.
2. For $0\le i\le m$, segment $P_iP_{i+1}$ is parallel to the $x$-axis if $i$ is even, and to the $y$-axis if $i$ is odd.
3. For every $0\le i\lt j\le m$, the segments $P_iP_{i+1}$ and $P_jP_{j+1}$ share at most one point.
Determine the maximum possible value of $m$.