So, British mathematician and physicist Roger Penrose has designed a chess puzzle that, he claims, is impossible to solve for a computer, but easy for a human being. Now, I don’t want to discuss the merits of the claim here (if there’s any interest, we could do so in Great Debates); rather, I’m merely interested in the puzzle’s solution.
Penrose claims that white can force a draw (which, to my layman’s eyes, seems obvious, as there doesn’t seem to be a way for black to win, barring catastrophic errors on white’s part), and even force black into a ‘blunder’ that allows a victory. But I’m not much of a chess buff; hence, I’d want to see the experts on the board discuss this puzzle. Is it as obvious as it seems, or am I missing something? Is it an interesting puzzle? Is there any special insight needed to fully appreciate it?
I suspect that the gimmick lies in the line “it is a legal position”. Right off the bat, it’s obvious that it’s extremely difficult to reach that position legally, and I wouldn’t be surprised if a more detailed analysis showed that it was impossible to reach it… if you assume that white is moving up the board as is typical for chess positions. But if you relax that assumption, the position might be legal. Except that a computer wouldn’t be programmed to consider the possibility that the board is upside-down.
The catch with this idea, though, is that
If black is moving down the board, his pieces are pretty well locked up, and so a draw is plausible, but if he’s moving up, then he’s about to promote a pawn. There’s nothing I can see White can do that can prevent that, and once done, it’d be very difficult for White to answer it.
And now off to read the article, to see what it says about it.
Supposedly I’m a chess expert , so I’ll have a go.
All Black’s pieces on the Queen-side (the left-hand side) are stuck - they have no legal moves.
The three Black bishops can only wander around without affecting the position.
White can draw simply by moving his King about.
After 50 moves by each side with:
no pawn moves
no captures
the position is declared drawn under the rules of chess.
I don’t believe White can win with best defence by Black.
White’s best chance is to move his King to d7 (easily done), then play pawn to c7.
This threatens pawn to c8 promoting to Queen giving mate (even if Black moves King to b7.)
However if one of the three Black bishops takes the pawn on c7, then Black is winning (since eventually his King will escape, freeing his Queen.)
If you replace the Black Queen by a Black Bishop (perhaps removing one of the other Bishops to make the position legally possible), then White does have a forced win:
I’m by no means a chess expert, but once I worked out the three black pieces in the open are bishops, not pawns (I stared at the hand-drawn diagram for ages before I saw the computer version) I could easily see white draws by
simply moving his king around the white squares for 50 moves - as long as he doesn’t push the pawn, there is no way for black to harm him as he can’t release his major pieces.
But I can’t see a way for white to win, unless
black moves all three bishops off the h3-c8 diagonal for a couple of moves, allowing white to get his king into a8 or b8 followed by promoting the pawn to queen or bishop for checkmate. I don’t see a way for white to trick black into doing this.
I think the point of the puzzle is that if you put the position into a computer as white, it will want to capture one of the rooks on offer, which allows black to mobilise his queen and then win easily. But I’m not sure if modern chess computers would actually do that.
OK, the article says nothing about the actual solution. And it has a better diagram, which makes it clearer that the loose black pieces are bishops. Looking at it more clearly, if we don’t make the assumption I spoilered, the problem is trivial: All white needs to do to draw is to spend 50 moves moving his king around the white squares. And in fact, it’s almost impossible for that to fail to happen: The only way it fails is if White moves his pawn and immediately loses it to a bishop, which would reset the 50 move counter, but then the count would just start right up again after that, at which point there’d be no way to stop it. The only way that game ends in anything other than a draw is if white moves his pawn and black lets him get away with, which no player above the level of a barely-experienced child would do.
Penrose’s claim is therefore absurd. If the computer has any sort of time controls on it at all, then it’ll eventually make each move, and thus eventually hit the 50-move rule, and thus achieve the draw. And if we’re counterfactually assuming a computer with no time constraints, then it’d eventually map out the entire tree and see the same thing. The only sense in which one can say that the computer doesn’t solve this problem is that it might not see the draw coming until it’s right on top of it, but no chess player ever sees the conclusion of a game until they’re right on top of it.
D’oh, I overlooked that White was in a position to capture a rook! Yeah, that’s a trap that a computer could certainly fall into.
No it won’t, unless White captures a rook. Black has no possible means of escape without relying on White.
And even with the possibility of the rook capture, I still don’t think that this is insoluble for a computer. A computer doesn’t need to see 50 moves into the future to win; it just needs to see far enough into the future to see the result of a rook capture. It’ll consider one of the rook captures first, follow that line to a black victory, and discard it. Then it’ll consider the other rook capture, follow that line to a black victory, and discard it, too. Then it’ll be forced to consider some other move, which means one of the correct ones, follow that line as far as its computational power and time limits will allow, and not find a mate. If the programming in the computer is at all competent, it’ll then take a line which is not a guaranteed loss over one that is, and so make one of the correct moves. Repeat that the next turn, and the turn after that, and so on, and eventually it hits the draw.
Yeah, I just tried on this on the free chess app on my phone. It “scored” black as being hugely ahead on piece value. Put on Autoplay, it ran through 50 moves of the king and bishops dancing around each other in less than a minute. No attempt to advance the pawn or take the rooks.
What, exactly, is Penrose’s claim? That computers have to play through it to see that it’s a draw, whereas humans look and see “all the bishops are on black squares, so the white pieces are untouchable” without playing through it? Is that not just a function of how we programme chess computers though, not some fundamental limitation?
My question is why would the black player be so stupid as to have three bishops on the same color? I assume this is only possible if black had two pawns promoted to bishop. Why not just promote them to queens instead?
Thanks for all of the answers—looks my intuition wasn’t widely off, at least.
I think that’s it, yes: human players rely on some special faculty to ‘intuit’ or recognize that there is no possible way that the game could result in anything but both simply dancing around each other, with no way to touch the others pieces, while ascertaining the same thing by an exhaustive search would be prohibitively expensive, computationally—basically, going through all possible sequences of 50 moves leading to a draw.
Again, yes, I think so, too: it’s simple (or ought to be) to write down a proof that demonstrates that white can always just dance around black, no matter what, and thus, force a draw; it’s just that chess programs typically aren’t set up to produce such proofs. But that only tells us something about the way we write chess programs, not about any intrinsic differences between human minds and computers.
Possibly your question is rhetorical, but for the record, you are right - this combination of pieces, let alone this position, would literally never occur in an actual game of chess. It is (according to Penrose) a legal position, insofar as white and black colluding on choice of moves could create this position within the standard rules of a game of chess, but it wouldn’t actually arise.
After White loses his pawn, the Black King is constantly threatening to escape to b7 (letting the Black Queen out next.)
The White King cannot keep guarding b7. It can stay on c6 or c8 for one move, but then has to move away.
Once the pawn is lost, it’s a forced win for White.
Turning to the point of the puzzle, I think Penrose wanted to create a position where White typically has about 6 moves (4 with King + 2 pawn captures) and Black has about 20 moves (all with the bishops.)
Given it takes 50 moves by each side to prove the draw, the computer would have to analyse 26 to the power of 50 positions.
I realise modern computers are powerful, but that’s still a lot of moves…
I should really learn to think more before saying anything about chess.
And I’m still not sure that Penrose’s point is a valid one. A computer programmed to play chess can play that position optimally, as proven by Stanislaus’ cell phone. A computer programmed to play chess maybe can’t prove that the draw is forced, but then, a computer programmed to play chess also can’t walk the dog, cook dinner, or generate music. We don’t expect it to do any of those things because they’re not what it’s programmed to do, but all of those are things that a computer can be programmed to do. Likewise, I expect that it’s possible to program a computer to prove chess outcomes: It’s just something that nobody’s ever bothered to do (though people have successfully programmed computers to generate geometry theorems).
Well, he’s good at physics, but he thinks that makes him an expert at neurology, artificial intelligence, computer science, etc. His book “The Emperor’s New Mind” was one of the dumbest difficult books I’ve ever read. He goes through all this quantum physics and ends up with, neurons are different than silicon chips, so AI isn’t possible.
I repeat that Penrose’s point was that every chess computer will evaluate the position as a win for Black, whereas any experienced player can see it’s a draw.
Yes, unless we hear directly from Penrose the default assumption should be there has been a fair bit of misreporting of whatever the original claim was.
, so I’ll have a go.
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