You inspired me to try again. And after years of being stuck on this round, I finally looked up the best way to play and won!!
I've spent years - literally, I got to this point as a teenager and never surpassed it - playing defensively. I'd start off with the obvious move and then try at random to make lines, but more focussed on blocking their lines. It worked well until this round and then failed miserably.
Having looked it up, I've finished the round in... ummm... ten minutes or so,