Demystifying Seirawan-Kasparov 1983 king and pawn endgame I am currently studying the Tarrasch Defense 1.d4 d5 2.c4 e6 3.Nc3 c5, and naturally I've stumbled upon many Kasparov games as he frequentloy and quite successfully employed this opening throughout his career. In particular, in the game Yasser Seirawan vs Garry Kasparov (1983), a remarkable endgame position transpires:

 [title "Seirawan-Kasparov 1983, position of interest after move 47.f5"]
 [fen "8/8/5p2/p1k2P1p/Pp2P2P/3K4/8/8 b - - 0 1"]
 [startflipped "0"]

 1...Kc6 2.Kc4 Kc7 3.Kd3 Kd7 4.Ke3 Kc6 5.Kd3 Kc5 6.Ke3 b3 7.Kd3 Kb4 8.e5 Ka3 0-1


Kasparov wins this game, but his technique goes very quickly beyond me and I'm struggling to understand the maneuvers that followed, particularly, black starts retreating towards the 7th rank and then returns to c5, and only then advances the b pawn after white has to make a move away from the  its current d3 square. My questions:


Did white miss any drawing chances in this position?
What was the plan underlying black's retreating king march?
Given this starting position, is one already able to recognize a winning pattern for black?


My attempt with this question is to learn more about how to reason about such endgame and to understand how Kasparov composed this (to me counterintuitive) plan.
In this game Kasparov is showing a perfect demonstration of the triangulation technique in order to Zugzwang the white king. 

To be in a Zugzwang means, any move loses or more generally, worsens your position, and one cannot simply pass the turn and maintain the position. 

In the diagrammed position, the key idea to spot is that white would be in Zugzwang if it was their turn to move. Let's see how by playing the position 
assuming it's white's turn again after 47.f5: 

 [title "(1) Same position as in the question, but white to move"]
 [fen "8/8/5p2/p1k2P1p/Pp2P2P/3K4/8/8 w - - 0 1"]

 1.Ke3 {only move giving us some resistance} (1.Kc2 Kd4 {e pawn falls and black wins}) (1.e5 fxe5 2.f6 Kd6 {black's in time and wins}) (1.Ke2 Kd4 2.Kf3 b3 {b pawn promotes}) b3 (1...Kc4 2.e5 {and white draws}) 2.Kd3 Kb4 3.e5 Ka3 {last key move, stopping white's king from preventing b2, and black promotes first and with check, which wins trivially.}


Having spotted the winning idea when it's white to move, next comes the question of how to achieve this Zugzwang position?
Essentially, black needs to find a way to pass the turn, that is, intentionally lose a tempo in order to have the same position but it being white's turn. This is called the triangulation technique. 



Before unraveling Kasparov's triangulation plan, let's have a typical triangulation exercise:

Triangulation in Kings and pawns endgame:

In the diagrammed position below, black has the opposition as it is white to move. One sign that triangulation might be possible to achieve is to realise that one king has fewer squares than the other one. Here, black has only 2 squares d7,c6 in order to stop the advance of the c pawn and shoulder the white king simultaneously, while its counterpart, the white king, has 3 squares d5,e5,d4. So to pass the turn, the triangulation goes as: 1.Ke5 Kc6 2.Kd4 Kd7 3.Kd5 and mission accomplished: white has the opposition now as it is black to move. The resulting  opposition is winning as black is in Zugzwang: all 4 legal moves Ke7, Ke8, Kd8, Kc8 are losing (convince yourself as an exercise). 





Kasparov's triangulation:

Similar to our previous example: black has to achieve the same position but with white to move, so we need to waste a tempo and pass the turn. This immediately calls for a triangulation scheme! Its feasibility here lies again in the fact that white's king has fewer squares than black's. See the diagram below: 



where green shows black's triangulation plan, red shows black's threats, and blue shows white's only available squares, since:


anything past the 4th row (e.g. white king chasing the black king) loses to the b pawn promoting, 
Kd4 constantly loses to Kd6 (black has opposition and thus, wins the e4 pawn), 
and finally descending to the 2nd rank (e.g. 1...Kc2) gives free access to the e5 square for black, which again loses trivially (as white cannot defend both the e4 pawn and stop the b pawn). 


Therefore, white only has the c4,e3 squares, i.e. fewer squares than black's king, which means the basic requirement for a successful triangulation is met, and we should be able to reach the desired position with white to move! 

Now let's see the plan in action (annotated diagram):

 [title "(3) Kasparov's triangulation and Zugzwang"]
 [fen "8/8/5p2/p1k2P1p/Pp2P2P/3K4/8/8 b - - 0 1"]
 [startflipped "0"]

 1...Kc6 2.Kc4 Kc7 3.Kd3 (3.Kd4 Kd6 {and black has the opposition and wins as Ke5 can no longer be prevented}) Kd7 {the Zugzwang is already achieved: Kc4 leads to Kc6 Kd4 Kd6 black winning, and Ke3 completes black's triangulation as we will see} 4.Ke3 Kc6 5.Kd3 Kc5 {and triangulation completed: same position we started off from but with white to move! Black wins, the Zugzwang situation was discussed in diagram (1).}




Final remarks:

What a beautiful endgame! Don't be discouraged by the difficulty of these endgames, nor misled by the simplified depiction of the triangulation scheme above. Endgames, in particular kings &amp; pawn ones, are known to be notoriously concrete and counterintuitive. Moreover, judging whether a successful triangulation is possible, is often far from being a simple matter. For example, in the above we constantly had to make sure all the other lines for white were indeed losing and that white's best bet lied in dancing along c4-d3-e3, and only then it made sense to compare which king has fewer squares! There's no singular-universally valid rule of thumb that tells us whether a triangulation leading to putting the opponent's king in Zugzwang exists or not. There are only signs, which need to be carefully considered.

The execution of these techniques, and the conversion of such endgames can never be achieved intuitively, one mis-step and all our advantage and efforts are out the window. For instance, in the game as you showed in your diagram, 8...Ka3 is in fact the only move that wins! To improve, focus on expanding your knowledge of endgame techniques (opposition, triangulation, Philidor/Lucena positions, etc), and in practical play, in order to recognise whether a combination exists, like in the Kasparov game, the mindset generally is: 


spot an idea (e.g. is it Zugzwang if I pass? like in the above game) 
calculate relevant variations (see if technically the found idea is achievable, e.g. in the above: can I achieve a successful triangulation?)
Evaluate the final positions in your calculations (like we did above for all the discussed losing variations)