This problem is from the $2019$ Pan-Africain Math Olympiad
A square is divided into $N^2$ equal smaller non-overlapping squares, where $N \geq 3$. We are given a broken line which passes through the centres of all the smaller squares (such a broken line may intersect itself). Show that it is possible to find a broken line composed of $4$ segments for $N = 3$
I cannot see how one would do this construction. First, notice that after the first move one can cover at each move at most $2$ other new points, thus at the first move he must cover $3$ points, otherwise he’ll cover at most $2\cdot 4=8<9$ à contradiction. Now, we can do casework which seems to lead to a contradiction, but this must not be true since this is a contest problem. Yet, I can’t see where I went wrong?
