Inside a unit square $n^2$ points are placed. Prove that there exists a broken line that passes through all these points and whose length does not exceed $2n$.
I've been trying to solve this problem for a long time without any luck. Any solutions/comments would be apppreciated.
Here is a path of length $2n+1$, perhaps you can fix it.
Start with a path that goes backwards and forth, $n$ times, like mowing a lawn.
When you get near one of the $n^2$ points, you will be within $1/{2n}$ of it, so you can pop out and fetch it with a detour of length at most $1/n$.