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H. G. Muller wrote on Fri, Feb 23 08:33 AM UTC in reply to Jean-Louis Cazaux from 06:14 AM:

Well, if you play with black the Applet should demonstrate it to you. The key is that the Bison has (say) to be on c4 with its King on b3 to checkmate on a1, and to force the bare King to step to a1 it had to attack c1 on the move before. So it has to move from c1 to c4 in two moves. (Actually a retrograde capture plus a normal non-capture, but since it is symmetric and non-divergent there is no need to distinguish the move types.) It can do that via f2 or f3, which means it must have to be on one of these in the mated-in-1 positions.

It could be that the King was forced to a1 earlier without the Bison being on f2 or f3, e.g. namely when the Bison is on any other square it could reach from c1 (so a4, b4, d4 or e4). The Bison then has two moves to reach f2/f3, as the next move Ka1-b1 is forced without the Bison's help. (And you could also play Kb3-c2 to move the mate to the other edge, and make d3 the target where the Bison could checkmate, after coming from b6/c6.) If you look at what a Bison can do in 2 moves, it is a lot. From any of its move targets to any other. That is W, D, H, WX, F, D, A, N, FX, NX, DY moves, amongst others. For the close-by targets there are only very few that it cannot reach in two moves. It particular, from f3 it could go to a4 (via d6), b4 (via e6), d4 (via g6) and e4 (via h6), so on all these squares it has mate in 3.

If a piece can make these final steps to force mate on a bare King that is in a corner, it can usually force mate from any position, if the board is not too large. What is too large is determined by how strong the piece is, and a Bison is pretty strong. (But it cannot force a King to the edge all by itself, such as a Centaur can, so at some size it will lose its mating potential.) For a piece with only large leaps, the board must also not be too small.

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