Monday, November 19, 2018

Stalemate ftw!?

Today, Magnus Carlsen and Fabiano Caruana are about to play their eighth game of the world chess championship match. So far, seven draws.

If all twelve regular games are drawn there will be a tiebreak match of faster games. But there's already a world championship in speed chess, we don't really need another one. There have been other suggestions, like Chess 960, but again that's a different game.

I guess it's fair to say that chess at the highest level has a bit of a problem. It's hard to get the general public excited over a game that goes on for hours, and almost always ends in a draw. By the way, let's throw in a link to a sketch.

There is a nice article about the tendency for draws by Lars Bo Hansen, "Carlsen vs Karjakin revisited". A claim that has often been made, and that is stated in Hansen's article as well, is that "at the core, chess is a drawn game".

But we don't really know that. The computers have shown that there is still room to take the game to a new level. And even if the best engines draw too when facing each other, at least it shows that the earlier conclusion about the inherent drawishness of chess based on human grandmaster games was not well founded. Maybe at a higher level still, White wins all the time.

An old idea is to abandon or modify the concept of stalemate. According to the stalemate rule, the game is drawn whenever the player to move has no legal move. Normally that player will still have free squares for their king, it's just that putting your king en prise is illegal.

But stalemate a bit illogical. You can't pass if you have a legal move, and so-called zugzwang is an important part of the game. Yet the ultimate form of zugzwang, where you can't move your king without putting it in check, lets you get away with a draw.

The stalemate rule is also a real hassle whenever you explain chess to beginners, because it means you have to reach a certain level of skill before you can meaningfully play a game at all. It would be much easier to teach chess to kids if the goal was simply to capture the opponent's king.

What if stalemate would count as a win?

Matt Bishop wrote an article about this idea a few years ago. He seems to suggest that stalemate would simply count as a win for the stalemating player. A counterargument is that this would take away a lot of the beauty of the game. Not just the beauty of tricky stalemate combinations, but also the beauty of forcing checkmate.

Even though few games actually end in stalemate, abandoning the stalemate rule completely (so that stalemating your opponent would count as a win) would change the game drastically. As was pointed out in an answer to a question on stackexchange, you would no longer have to get your king in front of the pawn in order to win with king and a pawn against a lonely king. This could make some games less interesting, because a lot of endgame technique would then suddenly not be necessary.

There is another option, which seems to have been mentioned already by Lucena around the year 1500, of counting stalemate as a "half-win". Stalemating your opponent could for instance give you 3/4 points, or 2/3. Or it could be counted as a pure tiebreak parameter.

This way, the game would only be "refined". Since checkmate would still be better than stalemate, all current endgame theory would still be valid. But in addition, we would get a whole new theory exploring when you can and when you can't stalemate your opponent.

Here is an "endgame study" that shows some of the basic technique that you would have to know in this form of chess:


White to play and force a stalemate!

White can actually force a stalemate in 14 moves. I will post the solution, but not immediately in order not to spoil it if someone wants to give it a try!

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Spolier alert!

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Alright, here's the solution:

1. Nc5! This is the only move that doesn't allow Black to escape! 1. - Kb2 2. Kd2 Ka2 (this move puts up the longest resistance, but there is also the trap 2. - Ka1 where White must not play 3. Kc2? but instead 3. Nd3 for instance). 3. Kc2. Again this is the only "winning" move: White has to take the opposition. 3. - Ka3 4. Kc3 Ka2 5. Nd3. White could repeat with 5. Kc2 but 5. Nd3 is the only move that makes progress. 5. - Ka3 6. Nb2! (Nc5 repeats) Ka2 7. Nc4! (Nd3 or Kc2 repeats) Ka1! (the most testing defense). 8. Kd2! Here White could also play 8. Kd3 which is equally good. But 8. Kc2? makes no progress: After 8. - Ka2 White would have to go back with 9. Kc3. Back to the main line: 8. - Kb1 9. Kd1/d2 Ka1 10. Kc1 Ka2 11. Kc2. By maneuvering the king into opposition, White has now reached this position with Black to move. Now it's easy: 11. - Ka1 12. Na3/d2/d6 Ka2 13. Nb1/b5 Ka1 14. Nc3 stalemate!

Notice that 1. Kd2? is refuted by 1. - Ka2! (not 1. - Kb2 2. Nc5 which would lead to the line above with a different move order). Now if 2. Nc5 then 2. - Kb2 and White is in zugzwang, while if 2. Kc3 then 2. - Kb1! (not 2. - Ka3? 3. Nc5) and White can't make any progress.

Just to record some more thoughts on this: It's quite easy to see that K+N cannot "in general" stalemate a bare king. First, a stalemate can only occur in a corner. Second, if the black king is outside the region of the six squares a1, a2, a3, b1, b2, c1, then it's impossible to force it into that region without allowing it to escape again. Up to symmetry, the only way to force the black king into that triangle is if it's on a4, the white king is on c4, and the white knight controls a5. Then Black has to play 1. - Ka3. In order not to let Black out again, White would now have to make a move with the knight to a square where it controls a4. Since it was previously on b7, c6 or b3, that square would have to be c5. After 2. Nc5 there would follow 2. - Kb2 3. Kd3 Kc1, and Black gets out of the critical region at c2 or d1.

This means that if the black king is anywhere in the middle of the board, White can never force a stalemate. But notice that 8 by 8 is actually the smallest board-size where the argument works! Any smaller and there might be a different situation where Black has to move into some critical triangle!

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