HMMT 2012 S7

For what positive integers $n$ do there exist functions $f, g:\{1,2, \ldots, n\} \rightarrow\{1,2, \ldots, n\}$ such that for each $1 \leq i \leq n$, either $f(g(i))=i$ or $g(f(i))=i$, but not both?