A fly in a closed rectangular box measuring $1\times2\times2$ starts at one corner, visits each of the other seven corners exactly once, and returns to its starting corner. To get from one corner to another it flies or crawls in a straight line. What is the minimum possible length of its path?

1. The path has $8$ straight pieces joining corners. The box has only four edges of length $1$ (the edges in one direction); every other distance between two corners is at least $2$.
2. So at most $4$ pieces have length $1$ and the other $4$ have length at least $2$: the total is at least $4\cdot1+4\cdot2=12$.
3. This is attained: with corners $(x,y,z)$, $0\le x\le1$, $0\le y,z\le2$, go $(0,0,0)\to(1,0,0)\to(1,2,0)\to(0,2,0)\to(0,2,2)\to(1,2,2)\to(1,0,2)\to(0,0,2)\to(0,0,0)$; its pieces have lengths $1,2,1,2,1,2,1,2$.
4. The minimum is $\boxed{12}$.