Which Gothic end-games are won?

Is it known which end-games are 'generally won' in Gothic Chess? I know that 3, 4, 5-men DTM tablebases have been calculated, but in my experience it is nearly impossible to see if an end-game is won or drawn from such tablebases. The statistics is usually completely dominated by quick tactical wins, mostly captures of a hanging piece on move 1. This completely masks the battle between the initially-present material. With DTC tablebases it is easier, as the quick tactical wins all have very low DTC, and can be discarded on that bases.

An end-game would be generally won if it would not help the weaker side if he had all his pieces close together, defending each other and the King, near the center of the board, even if he has the move (provided the opponent's pieces are not hanging). For example, KQKR is 'generally won' according to this criterion. The only way the Queen will not win this is if the Rook side can capture the Queen (or deliver checkmate) within 3 moves (check to drive the King in front of its Queen, second check, and then RxQ). I think KQKNN is similar.

It would be interesting to know which Gothic end-games are similarly won. I expect that KQKR would still qualify. But how about KCKR and KAKR? Will the Rook always go down, in a long-term battle?

The 2+2 with only noble pieces (Q,C,A) will most likely be extremely drawish, due to the relative ease whith which such pieces can keep perpetually checking. Nevertheless, there will be some quite lengthy tactical wins, where the side that has the first move keeps manoeuvring while checking until it can get the opponent's undefended piece in its aim. Even in KQKQ there are a few fairly lengthy wins, but with unequal pieces it is much easier, as they can be attacked from their 'blind angle'.

Of the 5-men, Q,C or A against N+N, N+B and B+B could also be very interesting. On 8x8 Q defeats any such combination of minor pieces, and I expect it to be the same on 10x8. From the very lengthy win of A vs B+N I suspect that the latter must be generally won: many of the positions along the win path do not look especially unfavorable for the B+N side at all.

Another very interesting question is if KBBKN would still be won on 10x8.

I just resolved some 10x8 endgames earlier this year because of a hard drive corruption and KBBKN was one of them. There is a mate in 91 for a longest win and 130,963,859 draws with white to move. Black to move has 195,553,178 draws.

it is a general draw with 51.3% of all positions drawn for white and 61.7% of all positions drawn for black. Fortunately Vortex knows exactly which are which and can head for the right one(s) if they come up

It would be interesting to recompute this statisic where the side to move has no capture, fork, forking check, skewer, or skewering check.

Again, if you want to look at the stats I collected, they are here:

http://www.gothicchess.com/db_5_stats.txt

I am not sure how you can conclude it is a general draw from these numbers, and in fact I have good reason you are wrong on this:

If I look to the 8x8 Chess tablebase statistics for the KBBKN end-game on the ICGA website, I find that with WTM 51.77% is draw, and 48.23% is won. So the 'uncorrected' win fraction is even a tiny bit lower than in Gothic.

Yet in 8x8 Chess KBBKN is totally won. If you interrogate this EGTB (e.g. on the shredderchess website), for any KBBKN position you will set up with WTM that will not trivially lose (or trade) one of the Bishops through a fork it will give you a winning move.

Or is the problem here that the ICGA website includes Bishops on the same color, and your Gothic EGTB not? With Bishops on the same color it is of course a trivial draw, (only helpmates possible), and without the Knight even a legal draw (no mates at all).

Well, I guess it depends on your definition of "general."

If more than half of them won't win even with the strong side to move, I think it's safe to say you have a high probability of ending with a drawn result than vice versa.

Can you even show one position that is draw?

white king on a1
white bishop on e2
white bishop on a5
black king on c2
black knight on c1

white to move and draw.

white king on j8
white bishop on h8
white bishop on j7
black king on g8
black knight on h6

white to move and draw.

Those are positions where the weak side has 2 threats though. Are there any fortress draws from any of these endings??

The mentioned positions are extremely special. My main reason for asking the question was not to get a draw position (they are easy enough to construct), but to know through how many positions Ed had to search though before he had found one that is draw. If the draw probability would be 53%,the first or the second random try would usually have been a 'hit'.

These positions are what I consider 'quick tactical wins/draws'. In position 2 you have a (safe) fork (on K+ to begin with, and thus will capture the Bishop for a Draw. The probability for such a position is less than 1%: even a centralized Knight covers only 8 squares = 10% of the board. The probability that a Bishop and the King are on this 10% is 20% x 10% = 2% (20% because there are 2 Bishops).

The other is a 4-ply combination, which is even rarer: in the initial position a Bishop is attacked (20% probability) and has to be retreated. Then the Knight can give a fork on K and other B. We just determined that the probability that this would be the case on a given square was <2%. As the Knight can go to at most 8 squares, it is <16%. So together with the 20% of the initial attack we are already down to below 3%.

But in most positions that satisfy this criterion, the Bishop evading the first threat will be able to retreat in such a way that it defends the square where the fork can be given. So positions where a 4-ply tactic of this kind is possible are not more common than ~0.5%.

I don't think that there are any fortress draws in this end-game (on 8x8 there are not). It is especially tactics of this kind that I first eliminate by discounting the positions with DTC<3. But in this games they make up only ~4% of the positions where the strong side is to move. (There is another set of ~2% where the King attacks one undefended Bishop, and the Knight attacks the other.) They are much more common when the weak side is to move, then the 20% probability that a Bishop is attacked by the Knight and can be taken is the dominant case.

So if you ever get in an end-game like this, and there is no two-ply tactic that loses you a Bishop, you can be 100% sure that it is won. This is why I would call it a generally won end-game.

But let me put the question in another way: would you consider KBBK generally won? Your stats say that with the strong side to move, most positions (50.5%) are draw!

I guess that means there are a vast majority of "senseless" positions that are solved as the endgame databases are generated. I was remotely aware of this when solving checkers engames, since about 50% of all positions had jumps for the side to move, and another 25% were jumps for the side who was not to move.

As humans we tend to naturally do all of the filtering you were talking about in your post. So we discard all of the "noise" and we focus on what remains. From my random sampling, most were long wins in the KBBKN endgame.

The endgame generator has some code in it to find endgames that match certain criteria, like no captures or checks, or wins of a certain length, etc. I can query it for some interesting draws once it is done its current set of tasks, which will be the end of January 2008.

Now that I was able to retrieve all my EGTB-generator sources from my laptop's deceased hard disk, I could do some tests with the Capablanca pieces. Unfortunately I can do that still only on an 8x8 board, but I do not expect this to make very much difference for most end-games. (KBBKN could be an exception.)

There is a pronounced difference between KAKBN on the one hand, and KCKBN, KQKBN on the other. This to the point where I would hesitate to say KAKBN is generally won. The number of positions that is not won with white to move is:

KQKBN: 3.3M
KCKBN: 5.5M
KAKBN: 28.1M

out of 228M pseudo-legal positions. (Only positions that are essentially different w.r.t. 4-fold symmetry are counted; for the true number of possitions all numbers have to be multiplied by 4.) Of this, about 70M-90M are won by King capture (so illegal); the number of legal positions wich are won with white to move in KAKBN is only 132M. The 28.1M is a significant fraction of that: about 21% is not won for the A side even if he has the move. This is not all due to quick tactical loss of the A, such as existing forks or skewers on K+A: the KQKBN engame should be even more sensitive to that (because in a Knight's fork on K+Q the N does not have to be defended), and has only 3.3 non-won positions. There is thus a pretty large number of fortress draws.

This can also be seen from the number of wins (for white) with black to move:
KQKBN: 112M
KCKBN: 110M
KAKBN: 53.3MM

Here the B+N side even has a ~50% chance (for not losing), while against C and Q it has virtually no chance at all. Also on 8x8 KAKBN can be a very lengthy win: upto 150 moves to conversion (and then you still have to win the KAKB or KAKN end-game).

But KAKBN is still a mostly won end-game.

KAKBB only has ~3% of won positions (for white) with black to move. With white to move about half the positions are won, but the number is similar to the number of positions where you can capture the King. So I assume that these are all positions where you can capture an undefended B. The fact that there are 1.5x more of those when the B are on opposite color (and so cannever defend each other) supports this. In general, having the two minor pieces defend each other seems the dominant factot in deciding if black can draw this end-game. This makes B+N the hardest case for the defender, as the unlike pieces can never defend each other. B+B with unlike B also has this problem, but, being sliders, both B can quickly seek shelter near there King, which for N is more difficult.

KCKBB against unlike B seems generally won: only 4% of the wtm pseudo-legal positions is not won. With btm the won fraction drops to ~40%, but this is not surprising as against KBB about 35% of the squares is covered by black pieces and having the white King there would be illegal. And for a similar number of positions the Chancellor would be up for grabs.

KCKBB with unlike bishops is more difficult: 29M wtm positions are draw now (of 112M pseudo-legal or 70M legal positions). With btm only 8% of the positions can be won by white. Presumably because the first thing black does, is have the Bishops defend each other.

KCKNN, like KQNN, is generally won (only 2.6M ~1% non-wins with wtm). With btm white wins in 56% of the pseudo-legal positions, which is pretty much anywhere where black cannot start by capturing K or C. (KNN cover about 27% of the board.)

KAKNN is even more difficult to win than KAKBN: only 16.5% of the pseudo-legal btm positions is won for white.

Note that KQKBB is even generally won, so apparently even if the B defend each other. This suggests the manner in which Q beats minor pieces is entirely different from the way A or C do it: K+Q simply drive the black King into a corner with checks, largely ignoring the other pieces that are busy defending each other. (Picking them up through a few checks as soon as they stop doing that). With C you need zugzwang to win against a bare King, and the minor pieces allow black to make his move with those when moving his King would get him in trouble. But for two Knights moving one of them means they no longer defend each other, while for like Bishops, you have moves that do not break the mutual defence. A is probably not capable of preventing the black King to seek shelter near its minor-piece fortress, where Q and C can easily cut off a King. So to win with A you have to keep checking the black King to get a shot at one of the minors before the minors defend each other.

Note that all this was done on an 8x8 board.

Interesting observations H.G.

Do you want my endgame generator for 10x8? Send me an email to GothicChessInfo@aol.com and I'll reply with everything you need.

Well, thanks for the offer. But I am going to write my own EGTB generator soon now anyway, and I will make sure it can also do 10x8. So I think it would be better to concentrate on that, rather than trying to learn decipher the format of Nalimov.

For answering some questions that came up on mating potential of fairy pieces on bigger boards, I converted my old EGTB generator to 16x16 boards, or even x even sub-sets thereof. By disabling diagonal symmetry, I can also do rectangular boards now.

(Beware: this is not my announced super-fast EGTB builder yet. In fact it is quite slow and memory consuming, because it always assumes a 16x16 board, part of which I then make inaccessible for smaller boards. But it means a 4-men (pawnless) ending takes 1GB, while for my 8x8 generator a 5-men took only 160 MB. And it still cannot do Pawns.)

In order to validate the results: Could you confirm that the following stats for K+A vs K (black to move) are correct for a 10x8 board, Ed?

Longest mate with white to move: w: Ka1, Ai1 b: Kc3

[edit] This should have been: Ah1

Hello H.G.

Your position is a mate in 20 moves according to my tablebases. I have a mate in 21 moves as being the longest.

That position:

White Archbishop: h1
White King: a1
Black King: c3

White to move and mate in 21 with 1. Aa8!

If you want to test your positions, I already have the tablesbases online and integrated into Gothic Vortex.

Do this:

1. Download http://www.GothicChess.com/vortex.zip
2. Once you extract the folder, make sure C:/vortex/tablebases is the path to the tablebases.
3. To test your tablebases are installed properly, go to the SPECIAL menu and select RANDOM 3-PIECE CHECKMATE. It should produce a White-to-move position, make a move, announce mate in the bottom of the GUI. You can then use "Start Autoplay" to have it play it out, or check the results with your own data.

Note: it is also very easy to edit the board and set up any position you want.

Let me know if this is of help.

These numbers are a 100% match!

But, as I mentioned, our longest wins do not agree. I have 4 positions that mate in 21 moves, and your longest is a mate in 20.