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01. Fundamentals
02. Winning Tactics
03. Spectacular Traps
04. Opening
05. Endgame
06. Draw
Appendix I: Laws
Appendix II: Variants
Resources
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5. How to Win in the Endgame
The inexperienced player is often at a loss to translate material advantage into final victory. Having no clear idea of how to proceed, he drifts in a way that can become very frustrating.
Where you are ahead in material, the guiding principle is reduction of forces. In endgames with Kings you bring this about by playing to force your opponent's Kings to the side of the board. In that position, their mobility will be reduced to a minimum, and you can enforce your will.
Two Kings against one is an easy win. Three Kings against two is fairly easy once you have the idea of building a protective bridge, using this as a technique to threaten exchanges.
Four Kings against three is a more difficult proposition, and here you will have your work cut out for you. Such endings make rewarding study because your playing skill will increase accordingly.
Another technique which can often be put to good use in King endings is that of giving up two Kings for one, resolving the position to an ending where you can force a quick win by bottling up the remaining King. Such opportunities occur more frequently than you might imagine.
In all these ways, then, you can force the win systematically and purposefully if you know your goal and are familiar with the techniques that must be used to achieve victory.
Practice these endings until they are second nature. You will frequently encounter these positions, or similar ones, which provide many satisfying opportunities to display your skill.
The inexperienced player often has trouble winning the endgame of two Kings against one King when the lone King is in the double corner. In the position of Diagram 61, White smokes out the hostile King by forming a protective "bridge."
BLACK WHITE
…… 14-9
1-5 10-14
The bridge is formed.
Diagram 61 (White to play and win)
BLACK
 |
WHITE
5-1 9-5
1-7 5-1
6-2 -------
Now it would be a waste of time for White to play 14—10, as Black has . . . 2—6 in reply. So:
------- 14-18!
2-7 18-15
7-2 15-11
And White wins.
In checkers we have five classic positions which turn up repeatedly in practical play. They are known respectively as First Position, Second Position, Third Position, Fourth Position, and Fifth Position. Because of their practical importance it is well worth studying them until you have thoroughly mastered them.
In First Position (Diagram 62), White has two Kings against one, as in the previous example. But in addition Black has an unmoved man on 3, and this, in the eyes of an inexperienced player, makes the win impossible, or at least exceedingly mysterious.
Actually the win unfolds with enchanting logic and economy of means. Black's object is of course to leave his man on 3 unmoved, for there no harm can befall him.
Hence, White first smokes the Black King out of the double corner (as in Diagram 61). The result is that White threatens to trap the Black King on a side square. To avoid this, Black must progressively play . . . 3—8, then . . . 8—12, then . . . 12— 16 etc.
Thus the Black man gradually arrives in the orbit of the White Kings, which, by economical play, can simultaneously menace the Black King and the Black man. Superior force must tell, and eventually Black must lose his man; thus White reduces the ending to the situation of Diagram 61 and wins easily.
To the uninitiated player, this may seem a lengthy process, but it is all rigorously worked out and proceeds like clockwork. In studying this ending, it is a good idea to play over the text continuation (columnar moves) first. This will give you a good grasp of the winning method. Once you have accomplished this, you can play over the whole ending again, this time studying the alternative possibilities. Only by examining these alternative possibilities will you arrive at a full appreciation of this beautiful endgame.
Diagram 62 (White to play and win)
BLACK
 |
WHITE
BLACK WHITE
-------- 18-23
32-28 23-27
28-32 19-23
Again White forms a protective bridge in order to force the Black King out of the double corner. If now . . . 3—7 or ... 3— 8, we simply arrive earlier at the text play.
BLACK WHITE
32-28 27-32
So Black's King must leave the double corner.
28-24 32-28
24-20 23-19!
Forcing Black's reply, for if ... 3—7 or ... 3—8, White replies 28—32 and Black's King is bottled up for good.
20-24 19-15
24-27 15-18!
Success! Since . . . 27—32 or ... 27—31 would now be answered by 18—23 bottling up the Black King, he must move his single man.
3-8 18-15
Forcing Black's next move, for if ... 27—23, then White replies 28—32, when Black must play . . . 8—12 (since he dare not move his King) leading into the text play after all.
8-13 15-18
Again leaving Black no choice, for if ... 27—31? White wins with 18—23.
12-16 28-32
27-24 18-15
If Black now plays . . . 16—19 White wins with 32—27!! for after . . . 24—31 White replies 15—24; . . . 31—26, 24—19 (or 24—27) and Black's King will soon be trapped. Here you have a brilliant example of winning simplification.
Another losing method is for Black now to play . . . 16—20 (from Diagram 63) when there follows 15—18; . . . 24—19, 32—28!; . . . 19—16 (or ... 20—24, 18—23!!; . . . 19—26, 28—19 and White wins), 18—23; . . . 16—12, 23—19; . . . 12—8 (if ... 20—24, 19—23 wins), 28—32; . . . 8—11, 32— 27; ... 11—8 (if ... 11—7, 19—15 wins for White), 27—23. White then continues 23—18 and eventually traps Black's King. Admittedly it requires patience to study these lines of play, but the effort is well worth it.
BLACK WHITE
24-28 15-11!
Diagram 63 (Black to play, White wins)
BLACK
 |
WHITE
Forcing another weakening advance. If Black now continues . . . 16—20, then 11—15; . . . 28—24, 32—28; . . . 24—27, 15—19; . . . 20—24, 19—16 (or 28—32) and White wins.
16-19 32-27
28-32 2 7-31
If Black tries . . . 19—24 now, White wins after 11—16; . . . 24—28, 16—20 (or 16—19).
32-28 11-16
19-24 16-19!
But not 16—20?? when Black plays . . . 24—27! or ... 28— 32! and wins! But after White's last move (forcing . . . 24—27) he wins the man on 24, remaining with a simple win as shown in Diagram 61.
Endgames of three Kings against two often baffle the inexperienced player, because he does not know the basic technique involved. This is to steer relentlessly for an exchange of King for King, winding up with an easy win of two Kings against one. Another way is to sacrifice the extra King, leading to a position which wins because you can trap the remaining King. This is nicely illustrated in Diagram 64, which at first sight appears difficult because the weaker side is ensconced in the double corner.
White's first move looks like a blunder. Actually it is a very strong move based on a clever finesse.
Diagram 64 (White to play and win)
BLACK
 |
WHITE
BLACK WHITE
--------- 16-19!
This looks wrong, for if Black plays . . . 27—24 White must reply 20—27, allowing the two-for-one shot . . . 32—16. But then White continues 28—24 and he must win, as he will quickly force Black's King to a side of the board.
27-31 20-24
If Black now plays . . . 31—26 White replies 19—23! forcing an exchange. Then, after . . . 26—19 White replies 24—15 winning easily. Another case of winning simplification.
32-27 28-32!
A sacrifice that wins at once.
27-20 19-24!
20-27 32-23
And wins, as Black's remaining King is bottled up.
Here is another setting of the three Kings versus two Kings theme which boils down to the same winning technique. However, the situation in Diagram 65 is more difficult in the sense that it takes White some time to arrive at a clearly winning situation.
Diagram 65 (White to play and win)
BLACK
 |
WHITE
White has a plausible try in 19—23, but then Black defends with . . . 28—32. (In that case White does best to return by 23—19, for if instead 23—26??, Black sets up a two-for-one shot with . . . 27—24!, 20—27; . . . 32—14!)
BLACK WHITE
…… 18-15!
White's job is of course to smoke the Black Kings out of the double corner. If Black now plays . . . 27—31, there follows 20—24; . . . 31—26, 15—18; . . . 26—31, 18—23; . . . 28— 32, 24—28 and White has taken the first step toward reaching the set-up of Diagram 64.
28-32 19-24
27-31 24-28
White has gotten a King into the double corner. If Black now tries . . . 31—26, then 15—18; . . . 26—31 (or . . . 32—27, 28—32 and wins), 18—23; . . . 31—27, 23—19 and White has a win as in the text continuation.
31-27 15-19!
Now we follow the winning method of Diagram 64. Thus . . . 27—24 leads to a White win after 20—27; . . . 32—16, 28—24 etc.
27-31 20-24
And now on ... 31—26, White wins by 19—23!
32-27 28-32!
27-20 19-24!
20-27 32-23
White wins. He has trapped Black's remaining King. Again simplification has been the key to the win.
In Diagram 66, White wins by forcing a position in which exchanges are inevitable. To do this, he must build a protective bridge which will prove effective against either Black King.
Diagram 66 (White to play and win)
BLACK
 |
WHITE
White's task seems difficult at first sight, as each Black King has access to a double corner. Nevertheless, White can starve his opponent out of good moves. Again he works by threatening to force exchanges.
BLACK WHITE
--------- 18-15!
The necessary preliminary to a later 14—10. If Black now plays . . . 6—1, we get the text play after 14—10 etc. On the other hand, . . . 6—2?? would be immediately disastrous because of the reply 14—10. Black can prolong his resistance only by hovering around the double corners.
BLACK WHITE
24-28 23-27!
Now Black dare not play . . . 28—32, for White forms a protective bridge with 15—18! forcing . . . 32—23 when 18—27 leaves White with the standard win of two Kings against one.
6-1 14-10!
If now . . . 1—5, White plays 27—24! forcing a winning exchange. Note the skillful use White makes of the protective bridge.
28-32 27-24!
Should Black try ... 32—28 we now get a beautiful win by 24—19! In that case White answers . . . 28—32 with 10—6!, while against . . . 1—5 he has 19—24! In either case he forces a winning exchange.
1-5 10-6
White is all set to answer . . . 32—28 with 6—10! once more forcing a winning exchange, thanks to his protective bridge.
5-1 24-19!
Once more forming the bridge. Black must play . . . 1—10, and White replies 15—6 with a clear win of two Kings against one. You will find it worth your while to master the technique of winning simplification.
In Diagram 67 (Second Position), material is even. Yet White can win because his position is very superior.
Diagram 67 (White to play and win)
BLACK
 |
WHITE
Black's men on 20 and 21 are both blocked. White, on the other hand, can arrange matters so that he crowns the men now on 30 and 27. Once he does that, he forces an exchange of Kings and wins very quickly.
Here again the winning process is long but very logical and clear-cut. Study the text play (columnar moves) first, and after you have grasped the winning idea, go back to Diagram 67 and consider the alternative possibilities.
................................. BLACK WHITE
------- 32-28
................................. 25-22 28-24
Now Black cannot play . . . 22—25, 24—19; . . . 25—22, 19—23; . . . 22—25?? for then 23—18 wins right off. Consequently theBlack King must head for 5 and 1.
22-18 24-19
18-22 19-15
22-17 15-18
17-13 18-22!
Completing the first phase—White prepares to advance his man on 30 to the crowning row without allowing . . . 21—25 etc. (Instead, 18—14 is futile because of ... 13—17 and White must still find the right winning method).
13-9 30-26
9-6 26-23
6-10 23-18
10-6 18-14
6-1 14-9
1-5 9-6
5-1 6-2(K)
Now White brings his new King to 19, still preventing the Black man on 20 from advancing after White's man on 27 starts for the crowning row.
1-5 2-6
5-1 6-10
1-5 10-15
5-9 15-19
9-14 27-23
At last this advance is feasible.
BLACK WHITE
14-10 23-18
10-6 18-14
6-1 14-9
1-5 9-6
5-1 6-2(K)
Completing another phase. Now White must exchange off Black's King.
Diagram 68 (Black to play, White wins)
BLACK
 |
WHITE
1-5 2-6
5-1 6-10
1-5 10-14
5-1 14-18
1-6 18-23
BLACK WHITE
6-11 23-27
10-14 19-23
14-9 23-18
9-6 18-14
6-1 14-9
1-5 --------
The only move Black has left. But now White forces the exchange.
------- 22-17!
5-14 17-10
21-25 --------
At last Black is able to crown this man, which seems to render all of White's previous work futile. But White will soon succeed in bottling up this new King.
------- 10-15
25-30(K) 15-19
30-26 27-32!
If now . . . 26—31 White plays 19—24! and after . . . 20— 27, 32—23 White wins on the spot.
26-22 19-24!
Forcing an exchange which leaves Black's King helpless.
20-27 32-23
And White wins in two more moves.
Diagram 69 shows another setting for an endgame of three Kings versus two. Again the problem is one of forcing the weaker side out of the double corner. Again the winning side makes use of the simplifying technique, which involves: (a) exchanging King for King, leading into a winning endgame of two Kings against one; or (b) sacrificing two Kings for one, in order to force a position that is even in material but won for White because he can trap the remaining Black King.
Diagram 69 (White to play and win)
BLACK
 |
WHITE
White aims to transfer his King from 23 to 24 in order to drive Black's King from 27 to 31, where the latter can be bottled up.
BLACK WHITE
-------- 23-19
32-28 18-15
Now Black has no choice, for if ... 27—32?? then 19—23 wins immediately for White.
BLACK WHITE
28-32 19-24
Forcing Black's reply, since after . . . 27—23 White can bring about an immediate exchange with 15—19.
27-31 24-28
So that if Black plays . . . 32—27, there follows 28—32; . . . 27—23, 15—19, again forcing exchanges.
And if Black tries . . . 31—26, then 15—18; . . . 26—31, 18—23; . . . 31—27, 23—19!; . . . 27—31, 20—24! and White wins by means of the text play.
31-27 15-19!
Offering the pretty sacrifice with which we are familiar, thus: . . . 27—24, 20—27; . . . 32—16 for now White plays 28—24 and wins quickly.
27-31 20-24!
Once more constructing a bridge—a valuable winning technique, as we have already seen.
The strength of this diagonal formation can be appreciated if Black now plays . . . 31—26, whereupon White's 19—23 forces a winning simplifying exchange.
.......... 32-27 ---------
Now one would think that White has nothing better than the humdrum 24—20, stamping his last move as a wasted move. Instead, White has the familiar winning sacrifice.
--------- 28-32!
27-20 19-24!
20-27 32-23
And White has won the game.
One of the most difficult endings to win is two Kings plus a single man against two Kings. In fact, there are many drawing situations arising such as Payne's Draw (Diagram 91) and Roger's Draw (Diagram 93).
A standard winning position of this type appears in Third Position (Diagram 70).
Diagram 70 (Black to play and win)
BLACK
 |
WHITE
Black's problem is to advance his single man to the King row, or, in the event that White maintains the blockade, to find some compensating advantage.
Once more, since the winning process is a lengthy one, it is advisable to go over the main play first and consider the alternatives later.
BLACK WHITE
13-9 22-18
Here 14—10? or 14—18? would be a blunder because of ... 9—14, forcing an exchange that wins easily for White.
................. 9-6 --------
If White now tries 18—15, Black's best course is ... 6—2! leading into the text line.
But on 18—15 it would be wrong to play . . . 6—1? for White replies 15—10! establishing the draw, for example . . . 21—25, 14—17; 25—30, 17—14 and Black can make no headway.
------- 18-22
........... 6-1! --------
But now this is the right move, for if instead . . . 6—2?, 14—10; . . . 5—9, 10—6; . . . 9—13, 6—10; . . . 21—17, 22—18 and White draws by the seesaw maneuver . . . 17—21, 18—22; . . . 21—17, 22—18; . . . 17—21, 18—22 etc.
After Black's last move, White is forced to give up the block
ade on Black's King at 21; for if 14—10 Black can advance
5 q
On the other hand, if White plays 14—18 at this stage, Black wins quickly by bringing his King from 1 to 19 in this fashion: . . . 5—9, 18—23; . . . 1—6, 23—26; . . . 6—10, 26—30; . . . 10—15, 30—26; . . . 15—19 (now it is only a matter of time until Black lifts the blockade
on his King at 21),
26—30; . . 19—23, 22—26; . . . 23—18 (the blockade is lifted!), 26—31; . . . 18—22, 31—27 (if 30—26 Black wins easily with . . . 9—13); . . . 21—17, 27—31; . . . 9—14 and Black soon gets a new King with an easy win.
Diagram 71 (White to play, Black wins)
BLACK
 |
WHITE
--------- 22-18
21-25 --------
Black immediately pops out of the blockade.
Should White try 14—17, he allows the man on 5 to advance, for example . . . 5—9, 17—21; . . . 9—14! (beautiful), 18—9; . . . 1—5 (the point!), 21—30; . . . 5—14 and Black wins
quickly by bottling up the White King. Thus if 30—26; . . . 14—18; etc.
Another possibility after 14—17; . . . 5—9 is 18—15; . . . 25—21, 17—22; . . . 21—17! with a fireworks finish, as after 22—6 Black makes a clean sweep with . . . 1—19.
BLACK WHITE
------- 18-15
1-6! 14-170
6-2! 17-14
25-22 15-10
22-26 --------
White cannot keep his King indefinitely on 14 (to blockade the man at 5). For example, if 10—15; . . . 26—23, 15—18; . . . 23—19, 18—22; . . . 19—15 and White's King at 14 must depart because of the threat to exchange by ... 15—10. (This explains Black's earlier . . . 1—6! and . . . 6—2!)
-------- 14-18
5-9 10-6
9-13 6-10
Now . . . 13—17?? would be a blunder because of White's "breeches" stratagem 18—22!
..... 26-31! -------
This threatens . . . 13—17.
-------- 10-14
Forced, to stop . . . 13—17. But now Black's King is freed from his imprisonment at 2.
31-27 ---------
This King has to swing around to cooperate with the other Black King.
-------- 18-22
27-23 -------
Contemplating this possibility: 14—10; . . . 23—19, 10—14; . . . 19—15, 14—9; . . . 15—10 and Black has reached the final text winning set-up.
------ 22-25
2-7...... -------
At this point Black must be constantly on his guard against Payne's Draw (Diagram 91).
------- 25-22
7-11 -------
Diagram 72 (White to play, Black wins)
BLACK
 |
WHITE
If White now plays 22—18 we get ... 23—27, 18—22; . . . 11—15, 22—26 (not 14—18? because of ... 13—17! and wins); . . . 27—24, 26—22; . . . 24—20, 22—26; . . . 20—16, 26—22; . . . 16—12 and Black wins by playing . . . 12—8 followed by ... 8—3. This forces the King on 14 to move because of the threat of ... 15—10.
BLACK WHITE
------- 22-25
11-15 25-22
23-27 22-26
27-24 26-22
24-20 22-26
20-16 26-22
16-12 -------
This King is heading toward 3. Black is following the procedure outlined in the previous note.
-------- 22-26
12-8 26-22
8-3...... -------
Threatening . . . 15—10 and thus forcing White's King on 14 to move.
But note that 14—17 is refuted by ... 15—18! Thus White's reply is forced.
------- 14-9
.......... 15-10 --------
This wins, for either White gives up his blockade by moving his King on 22 (allowing . . . 13—17 etc.) or we get 9—5; . . . 10—14, 5—1; . . . 3—7, 1—5; . . . 7—10, 5—1; . . . 10—15!, 1—5; . . . 14—18, 22—25; and now the man on 13 can advance . . . 13—17, reaching the King row fairly rapidly and giving Black three Kings to two for a win.
A situation with four Kings against three Kings often puzzles inexperienced players. The solution (as in Diagram 73) lies in forcing an exchange of Kings, thus reducing to the manageable ending of three Kings against two.
Diagram 73 (White to play and win)
BLACK
 |
WHITE
With his first move, White completes the bottling-up process for the Black Kings on 30 and 31. However, the remaining Black King in the double corner remains free. This creates a problem for White, but he can solve it.
BLACK WHITE
------- 20-24
Leaving Black with only one move.
BLACK WHITE
32-28 23-19
If Black now plays . . . 30—26, then 24—20 forcing an exchange that leaves White with three Kings against two.
28-32 24-28
Cutting down Black's activities in the double corner. If now . . . 32—27, 28—32 forces a winning exchange for White.
31-27 19-16
Setting up another opportunity for a winning exchange, thus: . . . 27—23, 28—24; . . . 32—28, 24—19; . . . 23—27, 22—26 etc.
27-31 16-20
This leaves Black with little choice, for if ... 32—27, 28— 32; ... 27—23, 20—24; and now, after . . . 30—26 or ... 23—26, Black is forced into a simplifying exchange that wins easily for White.
31-27 22-26!!
This leads to a neat two-for-one win, or, in any event, a winning exchange.
30-23 28-24
Now Black could prolong the agony—though in a losing fight—by forgetting about the loss of his King on 27 and playing . . . 23—18. Instead, Black tries to save the vulnerable King— and loses to a different simplifying maneuver.
27-31 24-27!
31-24 20-18
And White wins easily with two Kings against one.
In Diagram 74 we have a similar position, which in fact winds up with the same winning process.
Here too Black's resistance is based on his occupation of the double corner.
BLACK WHITE
----- 19-24
Note that if Black now plays . . . 30—26, he simply hastens his defeat, as White replies with a meaningless waiting move, keeping his Kings on 22 and 13 at their respective squares. Then Black must exchange, and White has an easy win with three Kings against two.
Diagram 74 (White to play and win)
BLACK
 |
WHITE
And if Black plays . . . 32—28, White simply plays 24—27 similarly forcing a winning exchange.
32-27 24-28
27-32 -------
If instead . . . 27—23, 28—32 and now ... 23—26 or ... 30—26 leads to a winning exchange for White, while if ... 23—19 (instead of ... 23—26) White wins with 32—27.
On the other hand, if ... 27—23, 28—32; . . . 31—27, 22—26! wins for White.
BLACK WHITE
------- 13-17!
A valuable waiting move. If Black plays . . . 30—26, White simply replies 17—13! (or 17—21!) and wins.
If Black tries . . . 32—27, 28—32; . . . 27—23, then 20—24 wins. For on ... 23—26 or ... 30—26 White still wins with 17—13 (or 17—21).
31-27 22-26!!
White wins as in the previous example.
30-23 28-24
And White wins, as ... 27—31 allows a two-for-one shot by 24—27; . . . 31—24, 20—18; and ... 23—18 allows White to play 24—31 with three Kings against two.
Similar, and yet in some ways different, is the situation we find in Fourth Position (Diagram 75). The play is so delicate that Black can win only if it is his turn to move. If White moves first (see the play following Diagram 89) the game is a draw!
Here again, since the weaker side has many alternative possibilities, it is advisable to play over the main line first before examining the alternative moves.
Diagram 75 (Black to play and win)
BLACK
 |
WHITE
Black has several possible methods of winning. One is to force an exchange of Kings. Another is to entice the advance of White's King on 30, in order to facilitate the advance of Black's man on 21 to the King row. Still another method, which turns up in several variations, is a neat sacrifice which forces either an exchange of Kings or a two-for-one win (in a situation which reminds us of the win based on Diagram 74).
BLACK WHITE
28-24 32-28
24-20 28-32
22-18 --------
Black's "retreat" is purely a matter of strategy. He is rearranging his forces for more effective operations.
Diagram 76 (White to play, Black wins)
BLACK
 |
WHITE
Right now White must watch his step, for if 32—28 Black forces a winning exchange with . . . 23—27.
On 31—26, however, the win is more involved. There follows . . . 23—19, 32—27; . . . 19—24, 27—32 (if 27—31?; . . . 20—16 wins); . . . 24—28, 26—31 (if 32—27; . . . 28—32 wins); . . 18—23, 31—27 (if 31—26; . . . 20—24 wins); . . . 23—26!, 30—23; . . . 28—24 and Black wins (27—31; . . . 24—27!), as we have seen in a previous position. (This motif turns up repeatedly in the following play.)
Again, if 31—26 (this is still from Diagram 76); ... 23—19, 32—27; . . . 19—24, 26—31; . . . 24—28, 31—26 (if 27—32; . . . 18—23 wins as in the previous variation); . . . 28—32 wins as above.
BLACK WHITE
--------- 31-27
23-19 ---------
If now 32—28; . . . 18—22, 28—32; . . . 19—24 (intending to answer 27—23 with . . . 24—19 forcing a winning exchange), 27_3l; . . . 24—28 (so that if 31—27 Black wins with . . . 22—26!, 30—23; . . . 28—24), 32—27; . . . 28—32, 27—23; . . . 20—24, 30—26; . . . 24—27! again forcing a winning exchange.
-------- 27-31
19-24 --------
And now again on 32—28 Black wins with . . . 24—27 forcing an exchange. (See Diagram 77.)
More involved is 31—26; . . . 24—28 with these possibilities:
- 32—27; . . . 28—32, 26—23; . . . 18—22, 27—31; . . . 20—24, 30—26 (if 23—26?? Black wins at once with . . . 22—18), . . . 24—27! forcing an exchange that wins for Black.
- 26—31; . . . 18—23. Then if 31—27; . . . 23—26!, 30—23; . . . 28—24 wins; or if 31—26; . . . 20—24 forces a winning exchange for Black.
Diagram 77 (White to play, Black wins)
BLACK
 |
WHITE
BLACK WHITE
------- 32-27
24-28 -------
The beginning of the end, for on 31—26 there follows . . . 28—32, 26—23; . . . 18—22 winning as in Variation I above.
-------- 27-32
18-23! -------
Setting the stage for the final coup. If now 31—26; . . . 20— 24 forces a winning exchange.
BLACK WHITE
-------- 31-27
23-26! -------
The long-prepared sacrifice.
-------- 30-23
28-24 ------
And Black wins the King on 27, for if 27—31; . . . 24—27, 31—24; . . . 20—18 etc. In either case Black will crown his remaining man and then win the standard ending of three Kings against two, or two Kings against one.
The great lesson to be learned from these endings is that with three Kings against two, or four Kings against three, your aim should be to exchange relentlessly.
A subsidiary technique which you can often apply is to sacrifice two Kings for one, winding up in an endgame which is even in material but which you win because you can trap the remaining hostile King.
Remember that when you are ahead in material, you must force your opponent's Kings to the side. There they will have the least mobility, enabling you to enforce your threats with the minimum of resistance to your plans.
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