Slowest won 4-men end-game?

[h.g.muller]

I recently encountered a 4-men end-game that was generally won, but so close to the edge that it could take upto 75 moves to force conversion. This was for King + Rook vs King + Waffle, where Waffle is a piece (occurring in the Clobberers army of Chess with Different Armies as the Knight substitute) that moves and captures one step orthogonally, or jumps two steps diagonally (WA in Betza notation). Usually short-range leapers with 8 move targets behave as other minors, and can draw against the Rook (e.g. Knight or "Woody Rook" (WD), which moves 1 step or jumps 2 steps orthogonally).

[h.g.muller]

The DTC of this end-game also has a very unusual multi-peaked distribution:



99.6% of all wtm positions is won, 66% of all btm positions lost. But 64% of all lost btm positions has a DTC of more than 50 moves.

[byakuugan]

How often does an endgame need to be winning in order to be considered a general win?

These endgames are also long wins
King + DNA vs. King + Knight
King + WD vs. King + Ferz

DNA = 2,0 or 2,1 or 2,2 leaper
WD = 1,0 or 2,0 leaper

[h.g.muller]

That is a hard question. The problem is that a quite large fraction of all possible positions with a certain material combination is not tactically quiet. And after the tactics has been played out, you are in a different end-game, which can easily have the opposite outcome from the one you are interested in. Humans would more or less automatically discount such positions. E.g. when you are in KN.KQ with Ne1 and black Kf3, Qc2 most people would feel swindled when you tried to sell them this as "a knight can sometimes draw against a queen".

It also depends a lot on who has the move (because usually tactics end favorable for him). The exact numbers also depends on how you count; my generator counts all 'non-broken' positions, i.e. positions which do not have two pieces on the same square, and does not make exceptions for illegal positions. So many of the positions, especially with the strong side to move, are King capture, and I count them as wins. As a Queen covers nearly half the board, K+Q vs something will have 50% of the positions won through K capture, even if it is KQ.KQ, (in which case a lot of captures of a hanging Q would be added too) which you would normally consider totally drawish.

In practice end-games you would consider generally won (for white) will have >99% of all wtm positions won. It depends a little on if there are powerful defenders that have a large chance to already have a fork or skewer on King + something they can capture to draw. Having only 80% wtm wins is already an indication the endgame is a hopeless draw. The 80% can be easily all King captures (which are printed explicitly by my generator, for reference) or capture of a hangin powerful defender (which in case of a single defender next to King is almost equal to that of K capture, as the chances the piece is defended by its King are only ~10%).

The btm stats vary a lot depending on the defensive material: in a generally won end-game captures of the white King (printed by the generator as W check) or capture/trade of a white attacker are usually enough to spoil the win, and especially if black has material next to the King, (meaning that white usually has more material scattered around) the chances he can do any of that on the first move are quite large. So 40-70% btm wins (white wins, that is) is quite normal for generally won end-games. For drawish end-games the number of btm wins usually drops to ~10%, representing the probability that white has a fork or skewer in the start position. This is why I let the generator by default calculate DTC, rather than DTM: the forks and skewers (or similar tactics) can be easily recognized in the table that splits out the btm wins by duration. The tactical wins all have quite short duration, and yu really should discard those. E.g. in KR.KQ the all positions where the Rook wins with the Q-side to move take 3 moves or less, showing they are all tactical wins.

As for the endgames you mention:

WD vs F seems indeed a general win, althugh the wtm win fraction is only ~96%. With weak pieces you sometimes have the problem that they are born in a place where they can be 'chased to doom' by the defending side. E.g. a Ferz is no match even for a bare King, once it is cut off from its own King for protection. So there are lots of positions in KFFFK that are not won because one of the F can be chased (and KFFK is a trivial draw). This is really tactics, but because the involved pieces are so slow, it is tactics that can take a number of moves comparable to the board dimensions. And it does not require very specific positioning of the pieces (as forks and skewers do), just that the pieces are in certain sectors, so there usually is a whole lot of them. My guess is that what you see here is the effect of chasing a WD that starts cut off from its King to doom by K+F that causes the missing 4%. The fact that you see >60% wins with btm, and that this number is not heavily dominated by tactical wins faster than 6 moves, also points to a general win.

What I do in cases of doubt is just set up a position that I consider quiet and positionally favorable for black, and then see if the EGT says it is a win. E.g. white: Kb1 WDa1 black Ke4 Fd4. If this is indicated as a win, you can be pretty sure the end-game is what humans would consider a certain win.

DNA vs. N has the typical signature of a generally won end-game, with 99.3% wtm wins. (I see you beat my K_R vs K+WA 4-men duration with that!) Here there is no question of chasing anything; the DNA is way too strong for that.

[byakuugan]

It would be pretty simple to write a code that will mathematically determine the number of moves until "checkmate" in a 2-man simulation of a King vs. Ferz situation, or a King vs. Wazir situation, but for them to be of any use in solving endgames, there would need to be more complicated codes that determine whether or not the locations of the other pieces will conflict with the checkmating sequence.
I imagine that it wouldn't be too hard to set up a user-friendly interface where you can set up the pieces on a board rather than having to type each piece location.

[h.g.muller]

Well, that interface already exists, and is called WinBoard. I use it all the time to run fairygen. It makes it like playing any other Chess program, except that it is mandatory to first set up a position with the right number and type of man. You can even use 'retract move' if you played a move that makes the DTC suddenly drop a lot.

What could be better is that it would actually react to setting up a position with other material in WinBoard by automatically starting a new calculation of the EGT for that material. Currently the only way to do that is to restart WinBoard, and change the engine command line in the combobox of the startup dialog to the new end-game.