# Solvable studies that computers fail to crack

Are there chess studies that require an outside of the box thinking that goes beyond the concrete calculation capabilities of today's engines (e.g. Stockfish)?

I have sometimes seen certain endgames that are known to be fortresses (drawn) but since the engine have depth limitations they cannot see all the way through and keep assessing them as winning for one side (for example giving +2 evaluation). But are there examples, to which we know the forced solution (checkmating line), but still the engines fail to solve?

• This is an interesting question. The intuitive answer would be to say no, because computers calculate more moves to a greater depth than any human can and therefore it would be difficult for a human to find a mate that the computer would be unable to find. Another thing to take into account is the number of moves the engine analyses in each position, since some engines are programmed to select moves to look into instead of considering all possibilities. Engines like stockfish use brute force, so it is unlikely that they would miss a move that a human can see.
– Aric
Feb 27, 2019 at 15:19
• In general studies that take many moves and have a high branching factor are difficult for engines. Even if the large variety of different moves seems trivial to humans, engines have to consider them all with minimax. EDIT - at least non-A0 engines... Feb 28, 2019 at 0:43

Here is an outrageous mate in 57 puzzle which none of the top 3 engines are able to solve:

``````[fen "8/6pp/5p2/k7/3p4/1Q2p3/3prpp1/3Kbqrb w - - 0 1"]

1.Qc4 Kb6 2.Qd5 h6 3.Qc4 Ka5 4.Qb3 h5 5.Qc4 Kb6 6.Qd5 h4 7.Qc4 Ka5 8.Qb3 h3
9.Qc4 Kb6 10.Qd5 h2 11.Qc4 Ka5 12.Qb3 g6 13.Qc4 Kb6 14.Qd5 g5 15.Qc4 Ka5
16.Qb3 f5 17.Qc4 Kb6 18.Qb4+ Kc6 19.Qa5 f4 20.Qe5 g4 21.Qf5 Kd6 22.Qb5 f3
23.Qf5 Kc6 24.Qg5 Kd6 25.Qb5 d3 26.Qf5 Kc6 27.Qa5 Kd6 28.Qb5 Kc7 29.Qa6 Kb8
30.Qb6+ Ka8 31.Qc7 g3 32.Qc8+ Ka7 33.Qc6 Kb8 34.Qa6 Kc7 35.Qb5 Kc8 36.Qb6 Kd7
37.Qc5 Kd8 38.Qc6 Ke7 39.Qd5 Ke8 40.Qd6 Kf7 41.Qe5 Kf8 42.Qe6 Kg7 43.Qf5 Kh8
44.Qg5 Kh7 45.Qe5 Kg8 46.Qf6 Kh7 47.Qf8 Kg6 48.Qe7 Kh6 49.Qf7 Kg5 50.Qe6 Kh5
51.Qe8+ Kg4 52.Qf8 Kg5 53.Qf7 Kg4 54.Qf6 Kh5 55.Qg7 Kh4 56.Qg6 Kh3 57.Qh5#
``````

Aside from the fact that the mating line is an absolute unit, an important reason why engines are unable to solve this is due to a technique called "Null move pruning".

Null move pruning assumes that playing no move is always worse than playing a move, which allows chess engines to search deeper by using this assumption to prune moves which are probably bad. However, because of this assumption, null move pruning is tactically vulnerable in zugzwang positions.

In the above position, black is pretty much perpetually in zugzwang, so null move pruning prevents most programs from being able to see how black's king is forced around the board. As a result, Houdini, Stockfish and Komodo all report draws.

• Awesome example! I liked both yours and phonon's answer, both captured a different aspect of the difficulty that some of these studies present for engines. Yours exemplifies how by design of the pruning engines may miss certain lines and the long sequence (57 moves) makes it only worse as the "number crunching" scales terribly for the engine, whereas a human can objectively see the end of the line and correctly assess it as winning! In Phonon's answer, I really liked the fact that the engines completely miss this very elegant bishop underpromotion coupled... Feb 28, 2019 at 12:04
• ... with the oh so precise `Ra1` maneuver and they end up opting for much more complicated-less-certain lines, when there's a much shorter forced win. Thanks to both of you, hope you don't mind that I accepted the other answer, the underpromotion I found really cool :). Feb 28, 2019 at 12:08

Indeed such studies do exist, similar to the fortresses you speak of. In my experience timely underpromotions seem to be the Achilles' heel of most current engines, and I remember at least one example that I'll show in the annotated diagram below.

It's a beautiful endgame study, with white to win (I strongly recommend trying it on your own before you see the solution here) and the difficulty lies in the fact that most lines that avoid stalemate end up in endgames that are in fact drawn (endless checks or impossible to make progress, at least as far as I remember) despite a large material imbalance (I'll show such lines as subvariations below). And since there's no immediate forced draw (by repetition or stalemate), the engine keeps evaluating the position wrongly and misses the winning solution.

I remember this study from a couple of years ago (I'll add the reference to the problem if I can find it), and I recall at the time the engines couldn't solve it (or maybe they needed days to see it, but I never let them run for that long):

Here's the FEN: `5K2/p2p4/p7/4P3/P7/8/1R4pp/6nk w - - 0 1`, to test whether an engine can solve it, simply give it the FEN and see whether it can find the critical line specially without you showing it our analysis and principal variation below (because once you do, the engine caches the line and remembers the solution thereafter within that analysis session).

`````` [title "White to play and win (don't remember the source at the moment)"]
[fen "5K2/p2p4/p7/4P3/P7/8/1R4pp/6nk w - - 0 1"]

1.Rb1 {stop knight moves and force black to exhaust remaining pawn moves} a5 2.Rd1 {Important waiting move with the rook, as there are no waiting moves with the king, as any king move to e-file blocks our own pawn, Kf7 blocks a key diagonal soon to be made clear, and Kg7 allows black to promote with check, for that see the added subvar} (2.Kg7 a6 3.Ra1 d5 4.e6 d4 5.e7 d3 6.e8=B d2 7.Bh5 d1=Q 8.Bxd1 Ne2 {and promotion's unavoidable, black's winning.}) a6 3.Ra1 {one of the critical moves missed by engine, at least not coupled with the right promotion, as to why a1 and not b1 for example, you'll see shortly} (3.Kf7 {a reasonable attempt would be to keep the rook on d1 and make a waiting move with king to force black's d pawn to move, and as such is what the engines also typically suggest but they end up in drawn endgames such as this very line} d5 {d6 leads to trivial mate as white promotes and black's still left with pawn moves to spare!} 4.e6 d4 5.e7 d3 6.e8=Q d2 7.Rxd2 Nf3 8.Rf2 g1=Q 9.Rxf3 {This position is not winning for white but the engine fails to see it due to calculation truncations and the overwhelming material imbalance is simply misleading it. Surprisingly, there are no ways to make progress, if my memory serves me well this is an actual fortress despite the engines giving it high centipawns.}) (3.Kg7 {alternative similar line to Kf7 that ends in a fortress as well} d5 4.e6 d4 5.e7 d3 6.e8=Q d2 7.Rxd2 Nf3 8.Rxg2 {else g pawn promotes with check and black wins.} Kxg2 {and black holds as white has no way to make progress, as the king cannot get close enough to overwhelm the knight. Again, surprisingly, there are no way to make progress, the fortress holds.}) d5 (3...d6 {leads to an even quicker loss for black as the d pawn's too slow and allows white a beautiful quick mate with a knight} 4.e6 d5 5.e7 d4 6.e8=N d3 7.Ng7 d2 8.Nh5 d1=Q 9.Ng3# {Therefore, d5 is black's only real attempt at salvaging the position.}) 4.e6 d4 5.e7 d3 6.e8=B {bishop underpromotion is the only way to win here! It's very simple once you see it, but otherwise everyone, including engines would opt for queen promotion without a second thought!} d2 7.Bh5 d1=Q 8.Bxd1 {right on time to blind the rook and prevent the stalemate} Nh3 {Ne2 no longer works for black as our king's on f8 not exposed to a queen promotion with check.} 9.Bc2+ Ng1 10.Bb1 {Bingo! that's why we needed Ra1 and not Rb1, the c2 square is needed to reroute the bishop to e4 with check while blocking the rook and giving black one more knight move to spare.} Nf3 11.Be4+ Ng1 12.Ra2 Nh3 13.Rxg2 Nf4 14.Rg3+ Ng2 15.Kg7 {waiting move} Kg1 16.Rxg2+ {and the win is trivial from here.}
``````

I hope this is close to what you expected. Some remarks:

• The lines with queen promotion lead to non-trivial draws that engines tend to fail to realise as the calculation truncation inherent to their working disables them from seeing a forced line that draws by force (say either by repetition or stalemate), moreover, the large material imbalance that is created with a queen promotion wrongly biases/fools the engine's evaluation such that it probably doesn't even consider a bishop promotion (as within the analysed depth it evaluates the position with the queen as favourable.) This is not to say that for humans it is easy to realise these fortresses, by no means, but after some time spent calculating few lines one starts to realise that objectively white has no means to make progress and convert. Eventually, one realises the bishop underpromotion is the only way to win.
• My guess is that if you give the initial position to an engine, it will opt for lines with rook on `d1` and followed by a queen promotion as its principal variation, and constantly evaluating the position wrongly with a large centipawn value (20+). Meaning they fail to see the fortress. However, once you show it the line with `3.Ra1` followed by the bishop promotion and `Bh5` it will see the forced checkmate and no longer reconsiders its previous attempt of `e8=Q.`
• For humans it is much easier to spot the solution as without much calculation one conceptually realises that stalemate has to be avoided when capturing black's promoted piece on d1, therefore a queen is out of the question. Moreover, the tedious part is rather connected to the lines with Rd1 plus the queen promotion which are more easily dismissed after a few tries as one eventually realises that despite being a piece up there's no way to make progress whether by winning black's queen or forcing checkmate. And once we've spotted the solution with `e8=B` the win becomes so trivial that we don't even need to consider alternative lines (even if they may still be winning).
• The fundamental difference between how humans tackle such problems (specially in endgames) compared to engines lies in our ability to foresee the end of a line conceptually without actually doing the concrete calculation, exactly similar to how we find fortresses in endgames but engines fail to see them. This allows us to contemplate alternative lines much more quickly than engines would do.
• A note for the line with `3.Kf7....9.Rxf3`, you can check that it's a fortress by removing the pawns on the a-file and asking the tablebase (with black to move), it will show you the draw. Otherwise keep trying with an engine and you'll see none of the lines lead to a conversion for white. Feb 27, 2019 at 16:16
• Wow! this is nuts...I'm trying the position using the cloud SF on lichess, and it completely fails to see the winning line you show and keeps saying +50 for the lines that are fortress! Neat!! Feb 27, 2019 at 16:19
• Interestingly, Houdini reports a mate in 31 for the `3.Kf7....9.Rxf3` line. Is there a strategy black can follow in that position that guarantees a draw? Feb 27, 2019 at 21:56
• Here is Houdini's mating line from the end of the `9. Rxf3`: `9...Qa7+ { #31/48 } 10.Ke6 Qb6+ 11.Kf5 Qb1+ 12.Kg5 Qc1+ 13.Kh5 Qb1 14.Qe5 Qd1 15.Qe4 Kg2 16.Qg4+ Kh1 17.Kh4 Qd8+ 18.Kg3 Qd4 19.Qh5 Qg7+ 20.Kh3 Qd7+ 21.Qf5 Qxf5+ 22.Rxf5 Kg1 23.Rg5+ Kh1 24.Kg3 Kg1 25.Re5 Kf1 26.Kxh2 Kf2 27.Re6 Kf3 28.Rxa6 Ke4 29.Rxa5 Kf3 30.Rb5 Ke3 31.a5 Kd3 32.Kg3 Kd2 33.a6 Kd3 34.Rg5 Kc4 35.a7 Kd4 36.a8=Q Kc3 37.Qc6+ Kd2 38.Rd5+ Ke2 39.Qe8+ Kf1 40.Rd1# *` I still need to check the rest of the Kf7 line Feb 27, 2019 at 22:22
• OK, `Kf7` is a win with a mate in 36, according to Houdini. This is the PV: `1.Kf7 { #36/60 } 1...d5 2.e6 d4 3.e7 d3 4.e8=Q d2 5.Rxd2 Nf3 6.Rd3 g1=Q 7.Rxf3 Qa7+ 8.Ke6 Qb6+ 9.Kf5 Qb1+ 10.Kg5 Qc1+ 11.Kh5 Qd1 12.Qe4 Kg1 13.Qg4+ Kh1 14.Kh4 Qd8+ 15.Kg3 Qd4 16.Qh5 Qg7+ 17.Kh3 Qd7+ 18.Qf5 Qxf5+ 19.Rxf5 Kg1 20.Rg5+ Kh1 21.Kg3 Kg1 22.Re5 Kf1 23.Kxh2 Kf2 24.Re6 Kf3 25.Rxa6 Ke2 26.Rxa5 Kd2 27.Rh5 Kd3 28.a5 Ke4 29.a6 Kf4 30.Ra5 Ke4 31.a7 Kd3 32.a8=Q Kc2 33.Rd5 Kb1 34.Qc6 Ka1 35.Rb5 Ka2 36.Qa6# *` Feb 27, 2019 at 23:06

This one I solved but stockfish 10 does not manage in a few minutes at least.

(all of it's lines seem to fail to some combination of Bf7, Bxh5 and ...g5 preventing Nf4, with pushing the e-pawn - at depth 27 it realises this and says +0.2)

• Spoiler: 1.Kd8 Bf7 2.Kxe7 Bxh5 3.Nf4 g6 4.Ne2+ King anywhere 5.Ng3 and the winning plan is to force black into stalemate with the bishop and king and knight by driving it towards the knight on g3. Mar 21, 2019 at 22:46
• Source : Kasparyan Domination Mar 21, 2019 at 22:47