Solutions for "Endgames To Make Your Head Hurt"

Some Discussion on Article 17

Article 17 of the Codex for Chess Composition is just a single sentence:

"Unless expressly stipulated, the 50 moves-rule does not apply to the solution of chess compositions except for retro-problems."

while the short version of the 50 Move Rule states: (Laws, Article 5.2b):

"The game may be drawn if each player has made at least the last 50 moves without the movement of any pawn and without any capture (see Article 9.3)."

So in real life, players can choose whether to claim a draw. The problem with the convention is that it doesn't tell us how to deal with this human decision element. Compare Article 18 of the Codex on Draw by Repetition:

"A position is considered as a draw if it can be proved that an identical position has occured three times in the proof game combined with the solution."

This is better. It replaces the Draw by Repetition rule - but if it does, then it should say so!

"3 Rep Revision 1: The rules for draw by repetition are replaced by the following. A position is considered as a draw if it can be proved that an identical position has occured three times in the proof game combined with the solution."

But it's still not ideal. It suggests that any triple repetition (yes I will call it this, even though arguably I should say "triple occurrence" or "double repetion") in the history of the game is enough the make the current position a draw. But that's not the way the rules work, and it leads to paradoxes. We should say that the

"(1) A position is considered as a draw if it can be proved that each player has made 50 moves without the movement of any pawn and without any capture in the proof game combined with the solution. (2) Unless expressly stipulated, this mechanism for draw applies only to retro-problems."

Article 9.3 is a complicated protocol by which the player with the move can optionally claim that either the last position or the next position will be claimable. But it conceals one important detail: what happens if the 50.0th move is checkmate itself? It's not stated in the Laws that checkmate trumps the 50 Move Rule. A Senior Arbiter confirms in real life that there are basically two cases:

  • Player is about to make the 50.0th move which checkmates but he could instead draw first. He chooses not to claim the draw because he's rational, and then he wins.
  • Player's opponent is about to make the 50.0th move which checkmates but the opponent could instead draw. It's not the player's turn so the game will end before he gets to claim the draw.

This seems intended to handle two types of composition:

  Example 1
Ken Thompson &
Peter Karrer

  (3+3) #262
  Example 2
Nikita Plaksin
500 Die Schwalbe 10 08/1971
618 FIDE Album 1971-1973
feenschach 30 10/1975
feenschach 38 04-06/1977
dedicated to Dr. K. Fabel
1.-2. Prize
  (15+14) #3

(1) Many "won" endgames have now been shown to require for completion a period of more than 50 moves between moves that "zero the counter" (pawn moves, captures or checkmate itself). Indeed, the FIDE Laws of Chess from 1993-1997 exempted certain classes of position from the 50 moves rule (50M). Although the Laws have since reverted, the Codex acknowledges in Article 17 that such endgames, which may be expressed as studies or compositions, should not be constrained by 50M. See Example 1, to take an extreme case. Solution here.

I can't resist repeating Tim Krabbé's awed comment:

"Playing over these moves is an eerie experience. They are not human; a grandmaster does not understand them any better than someone who has learned chess yesterday. The knights jump, the kings orbit, the sun goes down, and every move is the truth. It's like being revealed the Meaning of Life, but it's in Estonian."

(2) In a completely separate part of the forest, ingenious retrograde analysts have constructed many positions, where one can prove that a large number of moves have passed since the last pawn move or capture. Nikita Plaksin is the acknowledged specialist in this sub-genre, with a jaw-dropping total of 94 such entries to his credit in PDB. In some cases, 50M allows a solution to an otherwise impossible direct draw stipulation (e.g. =1). In other cases 50M renders a direct mate sound (as in Example 2) by removing duals. [1]

So basically Codex Article 17 allows these two groups (the "forward" endgame studiers and the "backward" retro-analysts) to happily co-exist. There are a few nits and gripes, but I'll get to those later.

How Do the New Compositions Fit in?

These compositions are based on a sneaky new way to abuse Article 17. Normally, in an endgame study, if one side has a forced loss in 262 moves, then that's it. But suppose that there are more than 50 moves to the next capture (or pawn move). And suppose that we can claim some element (even if relatively trivial) of retrograde analysis in the problem. Then after 50 moves a draw can be claimed under Article 17.

Well what is a "retro-problem"? It's one where the legality of the position is an integral element - i.e. if the composition simply wouldn't work if the pieces had been dropped on the board. Such a composition doesn't have to be as complicated as case (2) above. Almost [2] any use of the history of the game in deriving the solution is enough to classify a composition as "retro".

The easiest retro logic to accommodate is that determining who has the move. In a conventional study, White has the move, unless it can be proved otherwise. [3] So we want to set up a position where we can prove that Black has no possible last move. But it's not enough to just have White in check - we need something a little less mindless than that.

We then want to ensure that with Black to move first, any forced mate requires at some point >= 50 moves without pawn move or capture.

In the illegal position with White to move (and assuming) no 50 move rule, I prefer that the result would be different from the case where Black has the move. (So the retro logic really is essential to answering the stipulation "Who wins?") Also it makes the position more exciting.

To see how this works out in practice, we turn from philosophy to engineering.

These days [in 2004], all 5-unit and many 6-unit positions have been completely solved (by Ken Thompson, Lewis Stiller, Eugene Namilov, Christophe Wirth and others). Determinations of the "shortest route to win" for many pawnless positions are to be found in the pioneering online website "Play Chess with God", but as far as I know there are no plans for its further development. [indeed in 2017 it is long defunct.]

However, there are now other tablebases available on the internet (or in DVD form), and these can only increase in number and power as time goes on, although any one link might not endure.

Most current tablebases focus on DTM (Distance To Mate). However, since we are going to be interested in the 50 move rule, we need rather a DTZ (Distance To Zero) where Zero is defined as pawn move, capture or checkmate. A good introduction to the subject is A Guide to Endgames Tablebase.

I've just been checking the original Thompson results where I can against the multi-metric access set up by John Tamplin, and they seems to match well. Although they say it's experimental, the experimental aspect is in the front-end combination of DTM with DTC & other metrics, and not in the underlying database engine which is very robust. This is an excellent site, and I hope that it flourishes. The underlying code was written originally by Ernst Nalimov and then modified by Marc Bourzutschky, Peter Karrer and John Tamplin. It goes beyond Thompson's because it can switch Black and White and because it can specifically handle the consequences of the 50 move rule. There are some interesting papers on the site giving overall endgame statistics, lists of mutual zugzwangs, etc.

The user interface is currently not quite as convenient for composition as a Nalimov tablebase I found, where you can edit the board directly rather than having to copy/paste new FEN text each time. So for design I would use this Nalimov DTM, which does not understand multiple metrics to prune likely candidates, then check them out properly with DTC. But this is a minor point.

So after all that, now we turn to the solutions:

{A} This is the most complicated of the positions found.

(a) Logically, either side could have the move (consider R: 0. ... cxb1=B). So Article 17 never triggers and White has the move by default (Article 15). White can draw with 1. Kb4! Bc2/Bxd3 2. Kc3!/Nxd3! etc. 1. ... Bc2 2. Kc4? endures 197 moves before losing for White, as the 50 Move Rule does not apply here..

(b) Now Black has no last move. With Black to move, 0. ... Bf7+ has DTM=213, DTZ=197, and is winning for Black if not for Article 17. White can still draw by 1. Kg5! Kg8! 2. Kf6(Kf5) etc to hold Black at bay for at least 50 moves before the first capture.

This composition was published in the study magazine EG. Noam's article is reproduced here.

{B} In the hypothetical and unreachable situation where White did have the move, White has three mates in 1. But it's Black to move, and there are only two plausible choices. 1. ... Q×e4? 2. Ra1! draws but 1. ... Q×a6+! wins for Black but for Article 17. White has two sufficiently delaying moves 2. Ke6! (DTC 62 moves) or 2. Kd6! (DTC 57 moves).

If not for the 50 move rule, the solution would run: 0. ... Qxa6! 1. Bc7+! Ka7 2. Bd5! Qf6+! 3. Bc6! Ka6! 4. Bd8! etc. However the first alternative for White is at his 2nd move, so the solution stops there.

Black's first move threatens mate in 69, but alas for Black, DTZ is 69 too, so he cannot reset the counter by capturing a bishop on the way to mate, so White with proper play delays long enough to draw.

Note 1. ... Qc6+? or 1... Qd5+? show an uncommon kind of draw, forcing stalemate eventually by pushing wK backwards until capture of the desperado queen is the only choice. This kind of behaviour is more commonly seen by rooks.

It's perhaps odd that White, 2 bishops up, struggles even to draw with Black, whose pieces are bottled up in a corner, but that's part of the intended humour.

Without the 50 move rule, the strategy for the bishops' side in QvBB would be to set up the so-called "Lolli Position", discovered in 1763, a defensive fortress which can't be breached by the queen. (e.g. 8/1k6/1bb1Q3/8/1K6/8/8/8). In the current problem, Black is strong enough to avoid a fortress, but not strong enough to capture a bishop before 50 moves.

{C} White with the move would immediately mate in 1. But it's provably Black to move, with 0. ... a×b8=Q+! 1. Kc3! Q×g6! 2. Bc5! Qg3+! 3. Kc4! Qg8+! 4. Kd3! Qh7+! 5. Kc3! is the slowest win for Black with DTM=62 and DTZ=52 after 1. Qxg6. Article 17 therefore kicks in, and Black secures the draw at the 50 move mark because of it.

{D} White has the move. If he didn't, Black wins with 1... R×g1#. On the other hand, with White to move, his first move is forced but winning: 1. R×f1. Black can then move his knight anywhere but g1, 1. ... Nd4/Nh4+! are best with DTM=86 and DTZ=71, easily long enough to delay White's victory and claim the draw. So the result is draw.

[1] At this point, we stumble into a small hole in the Codex. Article 17 doesn't tell us how the players' decisions to claim or not to claim translate into the world of composition. The general convention is that if it can be proved that a total of 50 consecutive moves by each side occurred in the proof game together with the play, then a draw is automatic. I pretty much agree with that, with the rider that, like all the chess conventions, this one is only triggered when necessary to resolve ambiguity.

[2] If one side is in check, the position is meaningless with the other player to move, and you don't need the history of the game to tell you that. As far as the Laws are concerned, however, the only reason such a position is illegal is because it's unreachable. But, just about everyone would agree a check can't be enough to turn a problem into a retro-problem.

[3] In the case of studies (to win or draw), direct mate & selfmate, White has the first move by default. See Article 15 of the Codex for Chess Composition.

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