Optimal strategy in quantum tic-tac-toe

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Here’s something I discovered recently: quantum tic-tac-toe is a variant of tic-tac-toe which allows players to make multiple moves at once, in an attempt to simulate quantum entanglement and superposition. Apparently it was invented in part to provide a way of visualizing quantum concepts. In that respect, it seems to be a decent but imperfect conceptual aid, but it’s a pretty interesting game in its own right.

Anyway, tic-tac-toe is one of the simplest games there is, so the optimal sequence of plays have been known for a long time (in particular that if both players play optimally, the game always ends in a draw). But what about quantum tic-tac-toe? This question recently popped up on Board & Card Games Stack Exchange, and I’m rather curious to see what answers it comes up with. Currently it has a 100-point bounty attached, which means if you contribute the winning strategy, you could get 100 free reputation to get your start on Stack Exchange!