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.