TalkChess.com

[H.G.Muller]

Yes, there are exeptions. KBBKN is also often won. (Without the 50-move rule it would be a general win.)

But most of the time you need to be a major piece ahead in order to win, i.e. one that can deliver checkmate against a bare King. Such a piece can be quite a bit weaker than a Rook, though:

Of all fully symmetric pieces with 8 move targets the Knight is in practice the strongest. But there are two amongst those that do have mating potential, despite the fact that their middle-game value is less than a Knight: the non-royal King (a.k.a. Man), and the (1,0)+(2,0) leaper ('Captain' in Spartan Chess). I did some tablebases with those, and usually they seem good for a win.

1) KCN-KN -> won
2) KCN-KB -> won
3) KCB-KN -> won
4) KCB-KB -> won
5) KMN-KN -> won
6) KMN-KB -> won
7) KMB-KN -> won
8) KMB-KB -> won
9) KCE-KN -> won
10) KCE-KB -> won
11) KME-KN -> won
12) KME-KB -> won
13) KEE-KN -> draw
14) KME-KD -> won
15) KDE-KD -> won (but takes ~65 moves)
16) KME-KM -> mostly draw
17) KDE-KM -> mostly draw
18) KQ-KM -> mostly won, but...


Here E designates the (1,1)+(2,2) leaper, ('Modern Elephant'), the short-range equivalent to the Bishop. We see that in the end-game the E is slightly weaker than B: KEE-KN is generally drawn where KBBKN is a win. Yet in combination with the Captain B, N and E can all beat both N and B (like as well as unlike, when this arises).

The results 9-12 are relevant to Spartan Chess: the Lieutenant there is an E with a color-changing non-capture added to it. My tablebase generator cannot handle that, because this extra move breaks the 8-fold symmetry (the Lieuteant can only do it sideways). But since it is upward compatible with E, and end-game that is already won with E will certainly bewon with L instead. Similarly with the M: Spartan Chess has no M, but it has two Kings, and the Spartans can choose which of them they expose to check. so again this is upward compatible with having K+M. So it seems in Spartan Chess most combinations of light pieces can beat a single B or N.

Apparently having mating potential in the extra piece is the decisive factor here. Not so much that the pieces with mating potential are stronger. E.g. as defenders they are not necessarily better: KDEKD is generally won (albeit slowly), despite the fact that you are only a color-bound E ahead. Subduing KM is harder; M can be an excellent defender. This is evidenced by the fact that KQKM has a substantial number of draws. It seems mostly won, but it turns out that all positions where the K and M connect are in fact draws. It is only because the K and M are so easily forked by Q when they are far apart and undefended, and that it would take many moves to walk them towards each other, that the Q has so many wins. They are basically all tactical wins in 2 or 3 moves. The short-range D and E do not have the tremendous forking power of the Queen, so the defending M has much better opportunity to unite with its King.