The box problem that baffled the boffins

The intuition here that helped me understand this is that, if you know the search strategy of another player in advance, your best best is to "front run" where they will look as much as possible. So in the ideal case, look in the very next box they are going to look in each round. This guarantees that unless they guess right on the first round, you will get to the gift first.

The rows and columns thing is just a less perfect, but still useful, way for Andrew to front run Barbara's choices more often than the reverse happens.

One of those rare cases where Gemini Advanced outsmarts o1-preview. Gemini understood the importance of wasted turns when one player opens a box already checked by the other, and understood that this effect would penalize Barbara more than Andrew.

But of course these models are just stochastic parrots locked in a Chinese room. They don't "understand" anything, so never mind, nothing to see here.