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La lezione del GM SchwartzmanAs you hopefully remember from the last Lesson, Bronstein - Botvinnik was quite an exciting match almost half a century ago.  But no other game came even close to the level of excitement of the position you have been looking at for the past couple of days.   I would say that this is maybe the biggest blunder ever in the history of world championships.  But to understand what I am talking about, let me backtrack for a bit.
Please imagine this position one move earlier.  White's king was still on b3, and white was faced with a big problem, or rather two big problems: one on the 'e' file, and on the 'a' file...  Of course, to compensate for this big problem, or problems, white had an extra knight, something that should come in pretty handy, right?
Well, it certainly should have come in very handy. White could have checked with 1.Ne6, and then strategically placed it on d4, and it would have been an easy draw, as you can discover by calculating the few forced lines.
Instead, white decided to play 1.Kc2?? reaching the position we are looking at.  The idea behind this move is not very hard to grasp.  White's king is getting in closer, so that in case of the push of the 'e' pawn, he can quickly catch up to it on d2.  Further, if black were to play 1...Kf3, white still can catch up to the pawn with 2.Ne6, followed by 3.Nd4 and the elimination of the pawn in question, once again allowing for a draw to happen.
You might wonder if this is white's goal anyhow, then why go about it in such a hard fashion - why not play 1.Ne6 right away?  Well, there is a trick behind that.   You see, after the move in the game, if black were to go now forward with his 'a' pawn he might wake up to an unpleasant surprise:  2.Nb7!  By taking the pawn off the board, and then heading either with the knight to d3, and the king to the queen side, or depending on what black does, the king to e1 and the knight to b3, white actually manages to cover all bases, and win in a beautiful way!
So, white's idea was to set out what looked like a very safe trap:  black either falls for it by pushing either one of his pawns, or achieves a draw by playing 1...Kf3.   Sounds pretty good, doesn't it?
Well, it might sound good, but it certainly didn't look good.  Just imagine how Bronstein must have felt when black made his next move:  1...Kg3!!

What the heck is the black king doing there, you might wonder...   Well, here is the interesting story:  in strict mathematical geometry, the shortest line from point A to point B is a straight line - at least this is what I remember from school.  Well, let's just say that in chess, this simple rule doesn't help.  You could almost say that a chess player has to either get used to the geometrical rules of chess, or he should quickly forget what he learned in math class...
The reason I am saying this, is that in our particular case geometry is more important than you could ever imagine.  Just look at this situation:  the most important square on the board is e1 right now.  The 'e' pawn is the closest to promotion, and the place of that promotion happens to be indeed e1.  That is exactly where the white king is heading, and since he is already on c2, it is obvious that the black pawn doesn't stand a chance without the help of his king.  In order to be of help, black's king has to go to one square, and one only: f2.  That's the closest square that offers control over e1, so that is where the king has to stay in order to guarantee the promotion.
Unfortunately, as we have seen earlier if black takes the f3-f2 route with his king, he runs into a small problem:  the d4 square, and the fact that the arrival of the white knight there happens with a check, thus providing white with just enough time to catch the speeding 'e' pawn.  BUT, f3 is not the only way to get to f2.  Yes, it is a straight line, and yes, it looks like the fastest way there.  But in reality, g3-f2 is just as fast, and the important difference is that there is absolutely no check on d4 this time.  The result:  white is completely lost, and actually resigned right here...