Extra points for gnu plot and video!!! :)

But it's not solved yet: the challenge had three levels and you solved the easiest one.

And that to a very high cost. (75 min? Wow)

You seem to think that the vertices of the chain have to be at integer positions. (?)

That's not the case and crippling your solutions.

For the records:

Level 2: there is at least one solution with a real i.e. closed polygon! Or maybe more than one?

Level 3: what if the grid is bigger 5x5, 6x6 or even 6x5?

There doesn't seem to be much interest though and it might be a bit too challenging on the math level.