Suppose White has only a king and queen, and Black has only a king and pawn. Suppose further that the kings are connected or that Black can make them connected, since otherwise, White can mate Black easily with his queen. In this ending, Black's pawn is a liability. Recall that without the pawn (and with kings connected) the ending is drawn. However, with the pawn there, White can win by indirect explosion of the Black king.
If White can block the pawn to stop it from queening, then he can almost always win (there are very rare exceptions as we will see). The techniques for winning vary.
First, suppose that the pawn is sufficiently in the center of the board and that White can block it with his king before it advances too close to queening. For example, consider:
8/8/1k5Q/2K5/4p3/8/8/8 w
White to Move and Win
White begins by blocking the pawn: 1. Kd4+ Kc5 2. Ke3 Kd4. Next, White repositions his queen to attack the Black pawn: 3. Qh7 Kd3. Next, White runs away from the Black pawn, getting ready to catch the Black king in an explosion: 4. Kf4 Ke3 5. Kg5. White is threatening to capture the pawn and win, so the Black king is forced down: 5. ... Kf2 6. Qh3 Ke2 7. Qe3+ Kd1 8. Qd2#.
If we look back on this example again, we see that White had many choices as to where to reposition the queen to attack the pawn. Some squares are better than others. Let's try another:
8/8/1k5Q/2K5/4p3/8/8/8 w
White to Move and Win
1. Kd4+ Kc5 2. Ke3 Kd4 3. Qf4 Kd3. Now the White queen blocks the White king's path since 4. Kf3 is illegal and 4. Kf2 is met by 4. ... Ke2. So, White repositions his queen again: 4. Qe5 Ke2
8/8/8/4Q3/4p3/4K3/4k3/8 w
White to Move and Win
White now flees upward: 5. Kf4 Kf3 6. Kf5 Kf4 7. Ke6+. Now Black cannot safely follow with 7. ... Kf5 because of 8. Qxe4##. So, he will now get mated: 7. ... Kg4 8. Qf5+ Kg3 9. Qf3+ Kh4 10. Qg4#.
Let's tweak the last diagram a bit. Suppose we have this position:
8/4Q3/8/8/4p3/4K3/4k3/8 w
White to Move and Win
Here, directly fleeing upward with the king won't quite work. For example: 1. Kd4 Kd3 2. Ke5 Kd4 3. Kf6 Ke5 4. Kg7+ Kf6 etc. Black's king is ideally placed for the most flexibility. With White to move in the above diagram, we say that Black has the opposition. If Black's king were on d2 instead then White would win easily with 1. Kf4 Ke3 2. Kg5 etc. and if Black's king were on f2 instead then 1. Kd4 would easily win. There are two different remedies to this dilemma. White can waste a tempo with his queen to gain the opposition himself: 1. Qe8! Now, Black must commit to a side (left or right), and White will flee to the opposite side (staying in the middle with 1. ... Ke1 does no good because the kings get separated). For example: 1. ... Kf2 2. Kd4 Ke3 3. Kc5 Kd2 4. Kb6 Kc3 5. Qa4 Kd2 6. Qb3 Ke2 7. Qe3+ Kd1 8. Qd2#. Also, White can win without wasting a tempo to gain the opposition. He just needs to curl around the pawn like this: 1. Kd4 Kd3 (1. ... Ke3 2. Kf5 etc.) 2. Ke5 Kd4 3. Kf5 Ke5 4. Kg4+ Kf5 5. Qxe4##. In this last line, White could also have played 3. Kf4 and then if 3. ... Ke3, 4. Kg5 etc.
Sometimes, the White king cannot block the pawn in time unassisted, so the White queen does the job. Consider:
7Q/2K5/8/3k1p2/8/8/8/8 w
White to Move and Win
Here, White's king shouldn't chase down the pawn (1. Kc6 f4 2. Kc5 f3 3. Kd4 f2 etc.). Instead, White blocks it with the queen: 1. Qe5+ Kc6 2. Qf4. Now, there are many ways for White to win. For example, he can bring his king to f3, then switch blockers and proceed as in the above examples. However, this isn't always necessary. For example, 2. ... Kd6 3. Kd7 Ke7 4. Kd6 Kd7 5. Ke6 Ke7 6. Kf6 Kf7 7. Kg5 Kg6 8. Kh4 Kh5:
8/8/8/5p1k/5Q1K/8/8/8 w
White to Move and Win
9. Qg4!. Now, if 9. ... Kh6 White wins with some fancy footwork: 10. Qg5+ Kh5 11. Qf4! -- back to the diagram with Black to move and any of his three legal moves lose to 12. Qxf5##. So: 9. ... f4 10. Kh3+ Kh4 11. Qf3! -- just as above shifted down one row.
Now let's take a look at another example:
8/8/8/4Q3/1p6/8/2K5/3k4 w
White to Move and Win
1. Kb3 Kc2 2. Qb5 Kb2 3. Ka4 Kb3 4. Ka5 Ka4 5. Kb6+ Ka5 6. Qxb4##.
Now let's consider a pawn on the 7th blockaded. If White has it blocked with his king he always wins. It is not sufficient to attack the pawn with the queen from a distance, however. Once the White king slips out from its blockading position, the Black pawn must not be allowed to queen. So, the White queen will do double duty -- attacking the pawn and checking the enemy king. Consider:
8/8/5Q2/8/8/8/2kp4/3K4 w
White to Move and Win
1. Qc3 Kb1 (1. ... Kc1 2. Ke2+ Kd1 3. Kf3 Ke2 4. Qxd2##) 2. Qc2+ Kc1 3. Ke2+ Kd1 4. Kf3+ Ke2 5. Qxd2. It is always easy to win like this for a non-rook pawn. Let's look at a rook-pawn example. Consider:
8/8/8/7Q/8/8/5k1p/7K w
White to Move and Win
Since there is no room to the right of the pawn, White cannot win this like in the previous example. Instead, he employs a method strikingly similar to that used in "King and Two Queens vs. King." 1. Qd1 Kg3 2. Qf1 Kg2 3. Qg1 Kf3 4. Qg2+ and now the kings are separated and White mates with the queen.
Now let's consider the case where the pawn on the 7th rank is blockaded by the queen. Consider this position:
8/8/8/2k5/2K5/8/4p3/4Q3 w
White to Move, Drawn.
Black to Move, White Wins.
In this position, White's only chance at victory is to sweep his king down past the pawn and catch the Black king in an explosion (like Ke3-f2-g1 or Ke3-d2-c1).
Whoever is not to move has the opposition. If it's White's move, Black can remain as flexible as possible by staying on the square directly above White's king. For example: 1. Kd4 Kd5 2. Ke4 Ke5 3. Kf3 Kf4 4. Ke3 Ke4 5. Kd3 Kd4 6. Kc2 Kc3 7. Kd3 Kd4 8. Ke3 Ke4 9. Kf2 Kf3 10. Kg1 Kg2, and Black gets safely past his own pawn.
The only way that White can shake Black off of the square above him is to move his king to the top rank. For example: 11. Kf2 Kf3 12. Ke3 Ke4 13. Kf4 Kf5 14. Kg5 Kg6 15. Kh6 Kh7 16. Kg7 Kg8 17. Kf8 Kf7!
Black retains the opposition by moving underneath the White king. He will remain underneath the White king until White moves off the top rank, in which case, Black will resume his post on top of the White king. So, the position is drawn.
Let's go back to the starting position but with Black to play:
8/8/8/2k5/2K5/8/4p3/4Q3 b
White to Move, Drawn.
Black to Move, White Wins.
With Black to move, White has the opposition and White's idea is to let Black commit himself to one side of the pawn and then White rushes down the other way. For example: 1. ... Kd4 2. Kd3. White retains the opposition while moving toward the e-file. Notice that White can make errors (for example, 2. Kc3?? Kc4! and Black has recaptured the opposition for the draw). Play continues: 2. ... Ke4 3. Ke3.
Now, Black either plays 3. ... Ke5 or he must commit himself to the d-file or f-file. Let's look at both possibilities:
3. ... Ke5 4. Kf2 Kf4 5. Kg1 Ke4 6. Kh1! Kf4 7. Qg3+ and the queen mates.
Or: 3. ... Kf4 (3. ... Kd4 is similar) 4. Kd2 Ke3 5. Kc1 Ke4 6. Kb1 Kd4 7. Qc3+ and again the queen mates.
White's success above not only rested on the fact that he had the opposition, but also that the pawn was a central pawn (king or queen pawn). If the pawn is too near the edge of the board, White has no room for the above manouvering. For example, consider:
8/8/8/8/3k4/3K4/2p5/2Q5 b
Black to Move draws.
Here, even though White has the opposition, he can't win because there isn't enough room to the left of the pawn. Let's see: 1. ... Kc4 2. Kc3. Now, Black needs to be careful since 2. ... Kc5? loses to 3. Kd2 Kd4 4. Ke1 Kc4 5. Kf1 etc. and 2. ... Kb4? loses to 3. Kd2 Kc3 4. Ke1 Kc4 5. Kf1 etc. The correct way to proceed is of course 2. ... Kd4! and now White can try 3. Kb2 Kc3 4. Ka1 Kc4! and White can make no progress. Or White can try 3. Kc4 hoping to confuse Black, but that only leads to 3. ... Kc5 and Black has captured the opposition.
Knowing about the opposition here can help White make the correct decision with other king placements. For example, consider:
8/8/8/8/3k1K2/8/3p4/3Q4 w
White to Move and Win.
Here White can't simply run away from the Black king: 1. Kg3 Ke4 2. Kh2 Kf4 3. Kh1 Kg3 does no good. White can also go wrong with 1. Ke3? Ke4! and 1. Ke4? Ke5! both of which hand Black the opposition. Even 1. Ke5? Ke4 draws and we see that Black has an "underneath" opposition (2. Kd4 Kd5! for example). But White can win with 1. Kf3! Now, if Black moves to the e-file, White will grab the opposition (1. ... Ke4 2. Ke3! or 1. ... Ke5 2. Ke4!) and win. So, Black must try 1. ... Kd5 (since moving to the c-file is certainly suicide) 2. Kg2 Ke4 3. Qf3+! and the queen will mate.
So, far, we've covered every possible position of King and Queen vs. King and Pawn except rook pawns (not blockaded on the 7th). Let's look at those now
We've already covered the case where the rook-pawn is blockaded on the 7th. If it is not yet on the 7th rank, the technique is to force the pawn down the board to the 7th rank. Let's consider the worst-case scenario:
8/p1Q5/Kk6/8/8/8/8/8 w
White to Move and Win.
In our analysis, we will suppose that Black wants to advance his rook pawn as little as possible. If he advances it a lot, then White can blockade it further down the file and achieve a 7th-rank blockade position even faster. White's first goal is to put his king on a8 to achieve a set-up designed to force the Black pawn forward: 1. Kb7 Kc6 2. Ka8+ Kb7 3. Qb8. Now Black has no choice but to advance his pawn since if 3. ... Ka6 or 3. ... Kc6 then 4. Qb7+ and Black gets mated with the queen. Play continues: 3. ... a6 4. Ka7. Now if Black plays 4. ... Kb6 then White can play 5. Qb7 immediately forcing Black to advance the pawn again. So, let's say Black continues: 4. ... Ka8 5. Kb6+ Kb7 6. Ka5+ Kb6 7. Qb7 Kc5 8. Qc6+ Kb5. We have arrived at:
8/8/p1Q5/Kk6/8/8/8/8 w
White to Move and Win.
Notice that this position is the same as our starting position except that all pieces have been shifted down a row. So, White can repeat the process. However, since there are two squares above the pawn now, White has an even stronger threat. Let's see: 9. Kb6. Black must now advance his pawn already since if 9. ... Kc5 (or 9. ... Ka5) then 10. Ka7+ Kb6 11. Ka8+ and the Black king will get caught in an explosion. So, Black continues: 9. ... a5 10. Qc5. Now, 10. ... Ka4 loses immediately to 11. Qb5+. So, Black either advances his pawn or he continues with 10. ... Ka6 (10. ... Kc6 is similar) 11. Kb5. Now, Black must advance his pawn because of 11. ... Kb6 12. Ka4+ Kb5 13. Ka3+ and the Black king will explode. So: 11. ... a4 12. Kb4 Kb5 13. Ka3+ Kb4 14. Qc4 Kb3. We have arrived at:
8/8/8/8/p1Q5/Kk6/8/8 w
White to Move and Win.
Notice that this position is again the same as the starting position but shifted down some rows. So, play continues similarly: 15. Kb4 a3 16. Qc2. White's last move was designed to stop the pawn from advancing any further (he can't let it queen of course). Play continues: 16. ... Kc3 17. Qa2 Kc4 (17. ... Kb3 18. Ka5+ Kc3 19. Ka6 and Black gets mated) 18. Kb3 Kc3 19. Kb2 Kc2 20. Ka1+ Kb1 21. Qb3 Kc1 22. Qb2+ Kb1 23. Qc3:
8/8/8/8/8/p1Q5/8/Kk6 b
Black to Move, White Wins.
Now, Black must advance the pawn or explode and you've seen the rest before: 23. ... a2 24. Qc1 Kb2 25. Qb1 Kc3 26. Qb2+ and the Black king gets mated by the White queen in five more moves. Well, it was a 31-move mate from the starting position, but I hope you found the analysis logical and easy to remember. The point is to be able to remember the grand idea: force the pawn down the board to the seventh rank by threatening to explode the king.