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I imagine it's pretty powerful to put your queen on the corner of 4 squares so it can capture along two ranks and two files.

good point for sure, but in terms of relative power it's probably the same, as it could then be captured along two ranks, files, diagonals, and a fair number of knight, and even some pawn positions.

Yes, for example to capture two pieces at once, or threaten two adjacent pieces.

Indeed, or to bypass a piece which is threatening a space by moving into a different part of that space.

Any position where you'd be in zugzwang is a good example.

A more interesting question is how do you search the infinite state space. Intuitively, many states that are formally different aren't meaningfully different from each other and can be assimilated.

Any bisimulation expert on HN? :D