Could a polygonal path on $n \times n$ lattice cross itself?

Question

I am working on a problem from Problem solving strategies (Larson) and I encountered the following problem:

Let $S$ denote an $n$-by-$n$ lattice square, $n \ge 3$. Show that it is possible to draw a polygonal path consisting of $2n - 2$ segments which will pass through all of the $n^2$ lattice points of $S$.

This problem isn't supposed to be a simple induction problem. The goal, as I conclude from the chapter it pertains to, is to use induction along with generalization of the problem. In other words, I am supposed to prove a more general version of the problem (which I have to figure out) and solve it by induction. However, just poking around with some examples, I quickly arrived at a solution using induction if the paths are supposed to cross. I want to ask your opinion if polygonal paths are supposed to cross. I want to solve the problem and I don't want to cheat myself accidentally.

Answer 1

Here is a solution for 3x3 grid

It may show the induction for larger grids.