End-games: more armies

The Nutters

I adapted FairyGen to handle also two-fold symmetry (at the expense of the EGT being twice as large, and generation twice slower). This was a bit tricky, as this required distinction between retrograde and prograde moves, and flipping the orientation of the black pieces (neither of which was needed with 4-fold symmetry). But for 3-men EGT it finally gave identical results to the mating app here (which doesn't assume any symmetry). This means I could now do end-games with the Nutters majors as well. To keep everything together as an easy reference, I added the results to the tables in the previous comments.

The 4-men endings of light pieces were already interesting: it turns out the Charging Rook is very adept at beating other light pieces, much more so than an ordinary Rook. It has a general win against B, N, FAD, WA, and Fibnif single-handedly, while wins against BD and WD can in general be forced, but are then almost always cursed. I guess this success can be explained by that checkmating with Rook requires zugzwang, and will not work as long as the opponent has another piece to dump a tempo on. So you have to gain the other piece first, and in most cases this isn't any easier than checkmating (unless the additional piece is much weaker than a King, such as Ferz or Wazir), with the additional handicap that the piece can be protected by its King. Checkmating a bare King with the Charging Rook doesn't require zugzwang, however. So the mere possession of an un-involved defensive piece at a safe distance is no help. The piece must actively engage the Charging Rook, and the weaker pieces will perish in this combat. I did not calculate any 5-men EGT with Charging Rook + other vs defender where the Charging Rook would already beat the defender on its own; these should obviously be won as well.

A Charging Knight as defender behaves like a typical light piece: it loses against pairs of majors and (unlike) Bede/Fad pairs, and draws pairs of minors. Also for the Nutters, pairs of majors typically beat any single light piece. Apart from the WD the Charging Knight is the weakest major, though, and a pair of it has similar difficulties to beat a Rook, or its replacements Charging Rook and Dragon Horse. It does slightly better than the WD in this (as might be expected from the fact that it has one more move target), and has a cursed win against the Rook rather than a plain draw, etc.

The Nutters add new pairs of major + minor. These are interesting, because their ability to win depends on the possibility of the defender to choose which of the two pieces he will trade away. Charging Knight + Fibnif have similar difficulties here as Rook + Knight, against the Rook(-replacements) except Bede (which due to its color binding is apparently easy to dodge); the comparative weakness of Charging Knight compared to Rook is apparently compensated by the relative strength of the Fibnif that was already noticed before. Charging Rook + Fibnif does even better than Rook + FIDE minor, and beat almost anything, although its general wins against Rook or Charging Rook are partly cursed.

End-games with the Colonel are difficult to classify. Because of the extreme forwardness of this super-piece, the outcome will depend very much on where it is placed on the board. End-games where both players have a Colonel thus always contain a fair number of wins and losses, even if one would expect them to be draws. This even holds for the 4-men case Colonel vs Colonel: 15% of those are lost even when you have the move! (For comparison, for Queen vs Queen this is only 0.27%.) A Colonel beats most light pieces; it has mixed results against R, R4 and the charging Rook, while the Commoner (and thus the Dragon Horse) can hold a draw against it.

The Dragons

I also added the pieces from the Daring Dragons army: Commoner, vRsN (Dragonfly) and BW (Dragon Horse). This didn't really require any modification of the existing code for 4-fold symmetry, but to make a more accurate judgement on end-games containing more than one Dragonfly I put in some code to split the statistics in a 'like' and 'unlike' cases of the Dragonfly's special form of color-binding, and only report the result for the unlike pair here (as that is what you start with, and it is not a likely promotion choice).

The BW is quite strong, which should not be surprising, as its middle-game piece value is also slightly above that of a Rook. As a defender it can stand up to the Bede/Fad pairs in addition to pairs of other minors, probably because the Bede cannot easily sneak up on it from a diagonal, as it can against a Rook. (Note two FAD, which lack the distant diagonal attacks, can also not beat a Rook.) The BW is upward compatible with the Commoner, so in cases where a pair containing a Commoner already wins, replacing that Commoner with a BW should win even easier, and no EGT for these end-games were generated.

The Commoner (once under protection of its King) can keep a draw against a Queen and an Archbishop, because it cannot be approached by the enemy King. The Chancellor beats it, however. FairyGen cannot handle the ski-slide of the Wyvern yet.