George Duke wrote on Sun, Aug 23, 2009 08:43 PM UTC:
Actually though, can 2 quads per turn trap Rook or Bishop? Or under what
conditions? This gets as mathematical as it is strategic, and does depend
on the board sizes and what n is, n being #Rook or Bishop squares maximum
allowed per move. Understand there is no capturing either way, only
blocking, and that the piece starts from the center or one of the central four. It turns out from research of Conway and Gardner that board size
2n+2 sides (4n+3 for Bishop) with 2 quads per turn and n being that maximum
allowable move for either of these two pieces, traps them. That's why I was looking for the
sawtooth board in CVPage, and found it only at Gilman's Nested Chess and
related ones. Moreover, for a next step abstractly -- looking variously at 1,2,3 -- there are proofs incredibly that just one quad per turn is sufficient against Bishop or Rook now once again back to an infinite board. Strategy is to create corners, based on n, and work inward, progressively capping advance, if YOU ARE THE QUAD, even against some dramatically huge n. Result: certain eventual entrapment. If you have the chess piece, YOU'LL NEVER WALK AGAIN. Can Knight escape against one Quad on an infinite board? Unknown and unproved as of Gardner's writing twenty years ago. Now for CVs the interesting cases will be short-range pieces, rather than the sliders, on visualizable boards 12x12, 20x20, 30x30. I have not thought about these much yet, so the Rook/Bishop cases simply set a context for what would promise to be very extensive recreational-math treatment. Gardner does not take up any other pieces than K,R,B, and Conway in 'Winning Ways' only a few other for follow-up. Then we'll advance the subject to what we know now are more important CV pieces than British mathematician Conway could ever have dreamed of.
