A-ha, I knew it!
(Just kiddin', I had no idea :P)
The fact that every path passes through all the squares seemed something generative, but the levels were tricky enough to feel human-thought, so I was confused! But now it's clear, and I find your idea of computing a Hamiltonian path extremely intriguing - even if it may cause some computational difficulties, that however you overcome beautifully!
Thanks a lot for your answer, I loved to read this backstory! If it were for me, I'd present the game as "procedurally generated" in the first line: it's a fantastic plus! :D