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H. G. Muller wrote on Tue, Oct 11, 2016 10:23 AM UTC:

Color binding is a hard problem. For one, it does not only depend on piece moves, but also on the over-all board topology. On a cylinder board of odd width a Bishop is not color bound, and on a toroidal board with the proper helical pitch a Shogi Lance can access the entire board.

Even on regular Euclidean boards there can be surprises. E.g. the Chiral Knight (frbllfrbN) has a pretty severe meta-color binding, and can access only 1/5 of all squares. While in terms of normal colors one would have guessed it is a color alternator. In general, pieces that leap only in two bipolar directions cannot access all squares on a non-warped board, with the Wazir as only exception, because it makes the smallest possible step in both directions, and thus skips nothing.

BTW, color alternation is also a kind of binding, even more subtle than color binding, which imposes a conservation law on the evenness of the turn and square shade combined. This then prevents the piece to triangulate.


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