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And now for some good news. After playtesting Mir Chess, I have concluded that King and Cannon and Knight are strong enough to force a lone King into the corner and checkmate him.
For two Camels, two Knights or a Wildebeest we can prove that the longest forced mate is one move, because the geometry allows only a few mating patterns. This two-move example with two cannons is not trivial. Is it possible to prove that two moves is the maximum. Known is that the mating patterns is always with the Black King on the side (X1), White's King on X3 or on b3 against when X = a. And White's cannons are at Y1 and Z1 with Y between X and Z and Y not adjacent to Z. The last move is a vertical move by a cannon (C YA-Y1 or C ZA-Z1). Blacks last move is a horizontal King move, which can only be forced if the end file is involved (Second rank squares can only be covered by the White King), so this must be Ka1-b1, which means that X = b. One retromove by a cannon later, c1 must be covered. This is impossible. With two (Cannon + passive Bishop)-pieces (passive Bishop is a Bishop that does not capture), it should work (from: White CmB on c4 and c5, White King King b3, Black King b1, Black to move) 1. ...,Kb1-a1 2. CmB c4-f1, Ka1-b1 3. CmB c5-g1#. Probably (not certainly, it should be possible to force this with two CmB's and a King against a lone King. It would be interesting to prove this (and of course the King+Cannon+Knight against King)
I am not sure if King and Cannon and Alfil can force mate. Starting with White: King b4, Cannon b6, Alfil c5 and Black: King a2, how can we force the Black King back to the first rank, allowing the White King to move to b3? Note that the White Alfil cannot attack any square on the second rank.
perhaps slightly less than that of a Knight. When I replace two Knight by two Cannons in the FIDE array,
the Knights win about 52%, which is barey significant. (Giving Pawn odds would result in a 68% victory.)
Replacing the Bishops by Cannons leads to a 58% victory for the Bishops (over a 400-game match),
which is approximately what the Bishop pair wins over two Knights.
Using the Kaufman value N=325 (centiPawn), this means that Cannon = 320.
This makes it appreciably
stronger than the other well-known hopper, the Grasshopper, which has a value of 175 cP.
You are right about the mating potential of C+N and C+F. I adapted my
tablebase generator to handle
a Cannon. The mate with C+F apparently can only be enforced in te corner of the Ferz color,
and it migt take 59 moves to drive the bare King there. Cannon + Wazir is also a general win,
and goes slightly faster, as the mate can be performed in any corner (41
moves max).
Cannon+Knight is easy by comparizon: only 30 moves. With Cannon+Alfil it is almost always draw.
There are some mating sequences, though (0.5% with wtm, 0.1% btm). The longest takes 13 moves:
White: Kd6, Cc5, Aa7; black: Ka1 (wtm).
Staunton-style Cannon
Someone on wikipedia added the name Nao for a piece that moves like the Nightrider version of the Cannon. Does anyone know if this name actually was sourced from a chess variants book or was made up?
I've researched and I've found Nao multiple times in the Variant Chess magazine
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