Recently while teaching chess, the following position occurred in a tournament game that a student was showing me.

The student was Black and his opponent had just played 1.Nd5. He said "I think if I traded knights I would actually be losing." I doubted this, observing that after 1...Nxd5 2.cxd5 the squares f5 and e3 are "mined squares". Neither player wants to be the first to occupy theirs, since the position with the kings on f5 and e3 is mutual zugzwang. However, neither player is forced to ever occupy those squares, so the kings will dance around the e2/d2 and g5/g6/f6 squares, leading to a draw.

However, I soon realized that Black had a very nice and typical breakthrough:

Of course it is rare to find such an instructive and beautiful example from a student's game. I would be quite happy to get the opportunity to carry out such a combination in a real game. Unfortunately he did something else instead.

Of course, we have all seen examples of breakthrough sacrifices to create a passed pawn. For example, most people learn early on the following position:

But the breakthrough in the above position is really epic and stretches across the whole board. It is like invaders from space landing - as the defenders try to shoot each of them down they get overloaded until one finally lands and manages to plant the new alien crops.

It reminds me of this fantastic example from a game by Aron Nimzowitsch, which happened in a simultaneous exhibition held in Norway, 1921:

In the end it was Black's massive space advantage which had its say.

Finally we have two bizarre examples of a spectacular breakthrough. I say "bizarre" because by some weird coincidence the exact same combination was repeated twice in the early 1930s. First, there is the more famous example, which happened in Madrid in 1933. This was the game Ortueta - Sanz.

This spectacular combination became world famous. The entire game was actually annotated by Capablanca. However, in the actual game Black had additional pawns on g5 and e6 - which had been taken out when the above position was publicized. With the two extra pawns the black position was probably winning even without the combination, thus lessening its aesthetic value.

Much later the following combination came to light. It was played in Poznan, Poland, in 1931 - two years before the more famous Ortueta - Sanz game.

The entire game Tylkowski - Wojciechowski has also been published. It has been alleged that one of these positions did not actually occur. Some have said that the less-famous game from Poland was fabricated. It has also been said that the Ortueta - Sanz game was a fake. Personally I would guess that the Polish game was more likely to be real - precisely because it was less publicized (and also because it is known that the Ortueta-Sanz position was altered). However, both of these positions have entire games to go with them - with reasonable moves which stand up to analytical scrutiny. Believe it or not, it is extremely difficult to invent a reasonable, international-level game which reaches the desired position. Nobody really knows the answer to the mystery. Perhaps the stars were aligned the same way over Spain and Poland during those games.

Now it is your turn to solve some problems with pawn breakthroughs. In the following position, Black has just played the blunder ...g5??

The next one is from a game between Jose Capablanca and Edward Lasker (not the world champion Emanuel Lasker). Lasker had just played 40...K(d5)-e5, missing his chance to draw (which he could have gotten by 40...Ke6).

Finally I will show you the most absurd example of an epic breakthrough. The position would probably be unlikely to occur in a real game. All White has is his space advantage, but he can win by force. I won't put the solution here, but if you solve it you can leave a comment below.