Endgame Studies, Endgame Theory

Endgame theory for practical games is a part of the comprehensive endgame theory, it contains the positions that are most likely encountered in play. More exotic piece combinations might be relevant to both practical play and endgame studies, or only to endgame studies. The recently-proven win in the endgame of rook and two bishops on the same square color and pawn against queen and pawn (i.e. KRBBP-KQP with bishops on the same square color) that takes nearly 600 moves will never be relevant in a game, for example.

However, the example also provides the most important difference between practical endgame theory and such that applies only to endgame studies: The 50 move rule and all of its derivatives, such as the 75 move rule that existed for a time, are not in effect during the endgame study. Only the absolute truth in a position, not the practical outcome, is sought for. Human comprehension ends somewhere on the trenches of the battlefields of endgame tablebases generated endgame theory.

Marc Bourzutschky, 2021 (as reported in EG, October 2021). White wins (Distance to Conversion) in 584 moves

"Distance to Conversion" (DTC) is the metric that does not evaluate the length to checkmate ("Distance to Mate", DTM), which would be preferred by many endgame study composers, but to "conversion", which is the term for either a capture or a pawn promotion.

There also is a metric to evaluate the distance until the 50 move rule resets: "Distance to Zero" (DTZ). However, for the eight-piece endgames with blocked pawns only the DTC metric is currently available, and only due to the usage of 7 piece "endgame tablebases" (EGTB), which the game would become if a pawn is captured.

If the pawns weren't blocked, four more subsets of tablebases for the promotion of each pawn and 16 more subsets of tablebases for the promotion of both pawns would be necessary, all for eight pieces, i.e. a total of 24 endgames that don't need to be generated for the analysis of blocked pawns. (Andrew Buchanan sent me the following link with information from April 2021)

Thanks to Karsten Müller, I was able to ask Marc Bourzutschky. He replied (12 January 2021):

We have been focusing on the subset of 8-man endings that contain at least one set of opposing pawns, i.e., endings where there is at least one white pawn which is on the same file as a black pawn, on a rank below that black pawn. We call such endings “OP1”. A good half of 8-man endings that arise in practice are OP1. It takes about 1/100 the amount of resources to generate all the OP1 endings compared to generating all 8-man endings, so in terms of practical applications focusing on OP1 endings is a very good trade-off. Within the OP1 endings, we have more or less completed all endings with up to 4 pawns total that are of the form XPP vs YPP, or XPPP vs YP, where X and Y are non-pawns, as well as all OP1 endings that can arise from that set via promotion. Famous examples from the Karpov-Kasparov world championship matches are RBP vs BNP and NPPP vs BP.

This is just to give a short overview of where endgame theory currently stands: All endgames with up to seven pieces are known as well as some pawnless endgames with eight pieces and special cases of other endgames with eight pieces, such as with blocked pawns.

The book "Endgame Magic" by John Beasley and Timothy Whitworth has a helpful appendix that lists the endgames that are won. Again, this does not take the 50 moves rule into account, as it doesn't apply to endgame studies. The information there is the source for the following list. Bishops on the same square color aren't considered, if a player has multiple bishops. Please note that the evaluation of endgames would change on non-standard board sizes, for example KQ-KR was proven to be drawn if the chessboard has I think at least four times the usual size (16x16), as then the queen can't forcibly win the rook, as such the list only holds water on the normal 8x8 chessboard. We attach the important information from the list at the end of the article.

The endgame theory advancements led to corrections of seemingly incorrect studies or vice versa. The solvers of earlier times have found incorrectnesses that led to endgames that looked drawn, but were proven to be won by computers. Composers might have been certain that endgames were drawn, but they also were won, in which case a study might be incorrect. Jürgen Fleck's study that I showed a few months ago was a corrected version because...

Jürgen Fleck, "Schach" 01/1995, 1^st prize. White to move and draw (incorrect)

1.Kb7 attacks knight and pawn. It was thought that the game is drawn if White wins Pc7, and as such the author didn't consider 1.-Na5+, 1.-Nd8+ which de facto win for Black (reported by Martin Minski in EG, April 2015). The beautiful intended solution was 1.-Rf7 2.K:c7 Bg2+ 3.Kc5 Rf5+ 4.Kc4/Kd4 Rf4+ 5.Kc5 R:g4 6.Rh4 Re4 7.Rg4 Bf3 8.Rf4 Bh1 9.Rh4 Bg2 10.Rg4 Kb3 11.Rg3+ draws

Now I will first talk a bit about the history of endgames and then show some examples of tablebase knowledge being used in endgame studies.

As you will know, the term "endgame study" was coined by the 1851 book by Kling & Horwitz. I quickly talked about the history of endgame studies in January 2018.

Much more could be said, obviously, and an outstanding figure in the early 20^th century was Nikolay Grigoriev, who excessively explored pawn endgames (something that always remained en vogue, but he and later Mikhail Zinar almost exclusively discovered that terrain). Alexey Troitzky explored the endgame of two knights against one pawn exhaustively, creating the "Troitzky line", a pawn line spanning black pawns on a5, b6, c5, d4, e4, f5, g6, h5. This shows where two knights win if the single pawn is blocked there, and of course if the pawn is further back the knights also will win. This is because zugzwang is used to create a mating net, but the lone king will not be checkmated if the pawn hasn't enough useless moves.

The entire endgame theory there (in terms of what is the very best move each time) is difficult to understand, but the win is easy to practically execute, as it boils down to bringing the king to a corner with king and knight, then using the other knight also to checkmate it.

One fascinating thing about endgame theory is that it exists also for some crowded positions that aren't basic endgames, often because those or similar ones happened in games. Recently we saw one such example in Carlsen's sixth game against Nepomniachtchi when 17.-gxf6 was played because an endgame similar to the one after 17.-Qxf6 18.Qxf6 gxf6 had been in a world championship game of Carlsen against Karjakin, their first one in 2016. The first two replayable entries, found on page 52 and 53 of the book "Schachstudien der Weltmeister" by Karpov and Gik, demonstrate such pawn endgame theory. The "Magician from Riga" had a specific pawn formation in a game against Korchnoi, causing numerous grandmasters to analyze the position in the diagram below and reach a conclusion on how play should have went, creating a winning plan that was apparently unknown to two other players 17 years later.

Mikhail Tal - Viktor Korchnoi, Moscow 1968 (Candidates, semifinal). White to move only drew after 1.h3 Kf6 2.Kf4 e5+, etc.

Recently more relevant became the following position:

Black to move. Mamedjarov lost against Firouzja in 2021, but Alapin drew against Teichmann in 1908

Besides the practical endgame theory, there also is the more theoretical endgame theory, which can be used to debunk endgame studies. For example, He Sophie Yifeng on 19 January 2022 composed a small endgame study. Looking for predecessors, your author found an old Hildebrand study which she cooked immediately with tablebases. A "cook" is the term for an incorrectness. We reproduce the study and cook (this and Sophie's study are replayable below):

Alexander Hildebrand, Eskilstuna Kuriren 1957. White to move and win (cooked)

1.Nd5+ K:e7 2.N:c8+ Kd8 3.Nd6 leaves the rook in dire straits, two main variations follow:

3.-Re7 4.Ba5+ wins; 3.-Rg3 4.Ba5+ Ke7 5.Nf5+ wins

This is apparently forced, but 1.-Ke6!! draws, as the endgame of KBNN-KR is drawn. The rook will always attack the bishop, trying to leave White with KNN-K which is a draw. That the endgame is a general draw seemingly wasn't known in Hildebrand's time, but could likely have been analyzed, as the drawing idea isn't one that is beyond human comprehension.

What certainly is beyond the comprehension of some people is that Mario Garcia, I think, analyzed all endgame studies at the time with endgame tablebases (I think they had up to five pieces at the time) and checked if they were correct or not. Something similar yet has to be done with newer generations of tablebases and the endgame studies that came to exist since then.

While Emil Vlasák is the computer expert of EG (although to be fair all editors are computer experts to one degree or another), Jaroslav Polášek does work with endgame tablebases as well, his genuine efforts to correct and improve old endgame studies aren't restricted to tablebase testable studies, but a lot would be impossible without.

So to close this article, which is a bit shorter than usual but this is partially counterweighted with the appendix below, I want to demonstrate two examples of the work of my fellow EG writers, both taken from the October 2021 issue where also the tablebase endgame at the beginning of the article was found.

Both had investigated 8-piece endgames, Emil found a fatal flaw in a fortress by Klyatskin, which could have been found by human analysis long ago as well. The full study and winning way is replayable below.

M. Kliatskin, Shakhmaty v SSSR 1925 (2^nd semester). White to move and draw (end of study, cooked)

That both 3.Rc3 and 3.Rc1 end in the same fortress is not the big issue here, but rather the big issue is that the fortress after the intended 3.Rc1 Qd3+ 4.Ka1 Qc2 5.Rg1 (or any other sane move on the first rank) does not draw. I believe I had seen the position long ago before, and believed the author, but I am not fully certain of that. The easy winning idea is bKf2 (wanting to go to g2), Qd2. It seems that Paul Wiereyn, whom I quickly introduced in an article a few months ago as one of the computer programmers that are well-known in the specialized chess composition "scene", found the winning manoeuver already in February of 2021.

I use both the software and the chess composer Fritz regularly, and Jaroslav had found an idea that leads to of one of the ending ideas in the endgame studies of the Czechoslovakian master, funnily number 63636 in the database of Harold van der Heijden. The "humanly solvable" form is reproduced below, the longer version is replayable, taken from his own file

Jaroslav Polášek, after J. Fritz 1954. White to move and draw

The "Solving Championship of the Czech Republic" was held in Prague in August 2021, and this study was presented as one of the studies to solve. As such, I invite readers to try their hand on solving, prior to looking at the solution at the replayable studies.

Of course, this article can't cover much of how endgame theory in endgame studies is created or applied, how it always develops, but at the end of this article, an appendix must be made, the long following list, to show the current status of endgame theory in general. It might be a good reference, but can't replace individual calculations of endgames. As we saw with Alapin's play, precise knowledge is necessary of when pieces can be exchanged.

Appendix: Which endgames are won?

Pawnless endgames:

3 pieces:

• KR-K and KQ-K is won, everything else is a draw

4 pieces (3v1):

• KNN-K is a draw, all other are won

4 pieces (2v2):

• KQ wins against all except KQ
• KR might win against KN, if K and N are separated, generally it is a draw
• KR-KB is generally a draw, exceptions exist

5 pieces (4v1):

• All is won, even KNNN-K

5 pieces (3v2):

• KQQ-KQ is won, KQR-KQ is won, KQB-KQ and KQN-KQ are generally drawn, but exceptions exist which are shown in numerous endgame studies
• KRR-KR is won, KRB-KR and KRN-KR are generally drawn
• Two minor pieces against a rook or against one minor piece are drawn. Exception: KBB-KN is a win. There is a fortress for the KN player, but it was proven to be insufficient to draw by computers.
• The queen usually wins against two minor pieces, unless both are knights, where it is usually a draw if KNN are coordinated.
• KQ-KRR, KQ-KRB, KQ-KRN are drawn.

6 pieces (3v3):

• KRR versus K and two minor pieces is won.
• KRB-KNN is won.
• KRB-KBN is won if the bishops are on different colored squares, otherwise drawn.
• KRB-KBB draws.
• KRN-KNN generally draws, but the stronger side wins if he occupies the center, regardless of if KNN is far away from the edge.
• KRN-KBN, KRN-KBB are draws.
• KQB-KRR wins, despite it looking to be a draw at first sight. The strongest defense is to put the rooks on the sixth rank, but this still is won for the side with the queen.
• KQN-KRR is unclear. It might be won or drawn, depending on each individual position. The defense with the rooks on the sixth rank might be a fortress, but for example Ke6, Qb8, Ng7 - Kd3, Rd4, Re4 is won, so Beasley and Whitworth speculate that it is a win in general, and the fortress is an exception.
• While the book doesn't state it, KBB-KNN also is a general draw, the two endgame studies I showed above however prove that there is interesting play possible if the defender is in trouble.

6 pieces (4v2):

• KNNN-KN, KBNN-KB, KRNN-KR, KQNN-KQ are all won, despite KNN-K being a draw. The attacker as such must avoid exchanging pieces of equal value.
• KBBN-KR wins, but KBNN-KR draws.
• KRR and a minor piece against KQ wins, KR and two minor pieces against KQ draws.

7 pieces:

• KBBN-KBB and KBNN against K and two minor pieces are generally drawn. Otherwise the side with a minor piece or more up generally wins. Personally, I expect this trend to continue for 8 pieces and more, i.e. one minor piece up generally being a win.

Positions with pawns:

depend on details so no general rules can be given, apart from a minor piece up usually being enough material to win. There are exceptions such as KBP-K being drawn if the pawn is on a rook file and the bishop doesn't control the promotion square. KQ-KP is a draw with a pawn on a2 or c2 (or mirrored) supported by his king, unless the attacking king is close enough. This is due to stalemates in the corner that prevent the attacker from bringing his king closer. KQP-KQ and KQPP-KQ are highly dependent on where the pawns and pieces are, with both having winning potential. KRPP-KR is generally won unless the pawns are on the f- and h-file. Generally, positions with pawns have a higher probability of being won even with the 50 moves rule in effect, whereas pawnless positions might be theoretically won but practically drawn due to that rule.

Links:

• December 2021: Donnie, not Darko
• November 2021: Strong law of small numbers
• October 2021: Frankenstein and Monsters
• September 2021: "Schwalbe" - history tour of the German chess problem society
• All study of the month columns