Some Knightmate end-games

[h.g.muller]

I did some more Knightmate end-games. In this game the Knight is royal, and the Kings are normal pieces (aka Commoners or Men). There are no pieces with exotic moves here, so in principle these end-games can be generated with a generator for FIDE Chess. (I have one that does arbitrary pieces too, though.)

I had already found earlier that a Queen is the only piece with mating potential in this game. But then the mating is very easy indeed, as it does not even need support of its own Royal Knight; it just drives the bare Royal Knight with checks towards the edge, until it is mated. This makes N+Q vs N+something 4-men also very easy to win. The other pieces (R, B, K) don't have mating potential, but any pair of them does.
So:
NQN = won
NRN, NBN, NKN = draw
NQNR, NQNB, NQNK = won
NRRN, NBBN, NKKN, NRBN, NRKN, NBKN = won

I now did some 5-men. Of the 3+2 I found:
NBBNK = won
NKKNB = won
NBKNK = draw
NBKNB = won (both like and unlike Bishops)
NKKNK = draw
NBBNB = won

A defending K is apparently harder to beat than a defending B. Probably because you can avoid going on the color of thet Bishop, and still perform the mate with 3 against 1 on the other color. A defending Rook is in general a draw in such end-games, as the Rook can very easily keep the Royal Knight in perpetual check. In normal Chess a King can approach a checking Rook, and there is no safe check anymore when the King is close by. But checking a Knight is always safe for a Rook.

NBBNB is won when the white Bishops are on unlike color. (Otherwise it is draw, as NBBN is already draw with like Bishops.) It is a very lengthy end-game, it might take 104 moves to conversion.

For the 2+3 of course only those with white N+Q are of interest, as the others pieces have no mating potential even against a bare Royal Knight.
NQNKK, NQNKB, NQNBB are all won, the latter both with like and unlike Bishops.
NQNRK and NQNRB are mostly won (93% wtm, 35% btm, while the normal numbers for a won end-game are more like 97% and 55%). Presumably there are many positions where the Rook is engaged in a perpetual. B and K can never do perpetuals; against a B you obviously flee to the other color, against a K your Royal Knight out-runs it (if you do not stand mated in a corner).

NQNRR can go either way, there are about 1.5 times as many wins for the Q as for the RR (with the opponent to move), 15% vs 10%.