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Bob Greenwade wrote on Mon, Sep 18, 2023 10:12 PM UTC in reply to Ben Reiniger from 09:27 PM:

For starters, can two rooks force mate against a lone king? If the corners weren't removed I think they could. Two queens probably suffice? Maybe it's best to see if this has been resolved on the surface of a 3D cube first...

I don't think that two Rooks could force mate against a lone King without a board's edge to work with, whether it's a tesseract, cube, toroid, or whatever. Granted, the blanked-out corners might give a little extra leverage if this were a cube, but I'm pretty sure the split seams of the tesseract would always allow the King an escape.

On the other hand, two Queens could probably sandwich a King into the middle of a Face to force a mate -- at least a stalemate, if not checkmate. I think a Queen probably could do it with the Chancellor, though I'm not sure about the Archbishop.

I'd like to try to work out a bishop's path (and number of attacked squares) partly for this reason: it might be possible to effectively wrap around corners by going the long-long way around?...

Well, I can follow at least one path of a Bishop using the flat diagram: on an empty structure, the one that starts at 1.c5 can proceed to 1.c5, 2.d1, 2.e2, 4.a3, 4.b4, 4.c5, etc. (or the same numbering on Faces 7 and 9); or, continuing from 2.e2, to 14.c5, 14.b4, 14.a3; or, from 7.e2, to 14.e3, 14.d2, 14.c1...

Without mapping out the whole thing, I'm pretty sure just about any position allows a Bishop to cover eight of the 21 spaces on each side -- either a2, a4, b1, b5, d1, d5, e2, and e4; or a3, b2, b4, c1, c5, d2, d4, and e3 -- plus up to two additional spaces on the Face where it starts. The exception is the center square, where it's limited to one space in each direction.

"Wrapping the corner" would not be possible in a single move.

But I'm not sure that makes sense, in the same way that clockwise isn't unambiguous in 3D: it depends on from which side of the surface you're looking.

Well, there's an inside and an outside. I'm pretty sure the Faces are arranged so that they're all viewed from the tesseract's outside (I can check on that, though).


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