Saturday, January 06, 2007

A Proof For The Mousetrapper Conjecture?

Dan Heisman, in his recent article @ Chesscafe, has stated that checks always should be considered first, but given only if they do something good. His example of a bad check was from an endgame. I think this is no coincidence, which led me to the following

Mousetrapper Conjecture: «A check in the middlegame is more likely to be the best move than a check in the endgame.»

Common sense tells me that this should hold true. But I am not satisfied until I have managed to prove it by logic, derived from chess axioms. I use the following axiom list:
  1. The King must be safe from checkmate.
  2. A safe King must be active.
  3. A King can only be checkmated by pieces.
  4. The less pieces on the board, the less squares are controlled by pieces.
  5. A major piece giving check cannot block as many squares around the King as can block a major piece not giving check.
  6. The best move is the one which does most good for the player who is to move.
  7. What is good for one player is bad for his opponent, and vice versa.


Now let's begin with the logic. The King must be safe from checkmate. Checkmate can only be given by pieces. Therefore, with less pieces on the board, the King becomes safer. The definition on an endgame is not clear-cut, but generally, there are less pieces in an endgame than in a middlegame. Less pieces is equal to less squares controlled by pieces. A King is safer if he has access to more squares not controlled by (opponent) pieces. Therefore, a King is safer in the endgame than in the middlegame.

This also means that a King in check generally has more options in the endgame than in the middlegame. Therefore, his chance of getting a better (more active) position when escaping check in the endgame is likely to be higher than escaping a check in the middlegame. Hence a check in the endgame is more likely to be a bad move, chasing the opponent King to a better square. If the check is given by a major piece, the number of squares where the King cannot go is diminished, hence his chance is higher to get a better square, which is good for him, which is bad for the player giving check.

Quod erat demonstrandum. Maybe I should call it now «Mousetrapper's Rule»? BTW there exists also a practical proof: Try to checkmate a naked King with your Queen and King, and you will use more moves by giving multiple queen checks than by silently chasing the King in Knight's distance just to give only one final checkmate.

But the most important question remains: Will it help in a practical game? I think it could, because it is a good option to always consider checks first, and then comes the critical evaluation of whether the check is a checkmate, leading to checkmate, improving the position, gaining material, or if it does nothing, or if it helps the opponent. So, it might be no bad strategy: If you see a check in the middlegame, consider it at least once, if you see a check in the endgame, consider it at least twice.

4 Comments:

At 2:35 PM, Blogger Mousetrapper said...

(Received per Email from Dan Heisman)

Sort of reminds me of my comment in the NN "Learning from Dr. DeGroot" about how intermediate players sometimes reject captures just because they do not win material...

Regards, Dan H

 
At 2:44 PM, Blogger Mousetrapper said...

Very important point, Dan.

For a long time, I have been very hesitant about trading pieces in the opening, just because I did not want «to give my army away for nothing».

Later I began to realize that pieces are not of equal value, for instance, I get something if I trade my knight that has done nothing yet against a mighty knight in which the opponent has invested more tempi. I remember a number of openings where I got upperhand against stronger opponents after such a trade.

 
At 4:21 PM, Blogger Temposchlucker said...

Your theoretical reasoning is impressive. But I still have trouble to imagine how it can be made applicable in practice.

 
At 6:00 PM, Blogger Mousetrapper said...

Tempo, not much, I must admit. If you are used to check and doublecheck each check (nice prase, hehe) then a rule about the probability that a check is good or bad will add no help to this decision. But, still, it cannot be denied that a preparedness to be more critical on checks in the endgame than in the middlegame can be beneficial, theoretically. But my impression is that such reasoning at home can possibly add to general chess insight which in turn should help to find better moves OTB. If not, then chess reasoning is still a nice hobby.

 

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