This position, being of frequent occurrence, is most important, and I recommend it as a valuable study in the use of the opposition.

Before I discuss positions of greater complexity, in which the only way to win is by sacrificing the extra pawn, I shall treat of end-games in which positional advantages ensure the victory although the pawns are equal.  Here we shall find simple cases in which pawn manoeuvres bring about the win, and more intricate ones in which King moves are the deciding factor.

Of the former the most important type is the end-game with the “distant passed pawn.”  A typical example is the position in Diagram 58, in which Black wins.

---------------------------------------
8 |    |    |    |    |    |    |    | #K |
|---------------------------------------|
7 |    |    |    |    |    |    |    | #P |
|---------------------------------------|
6 | #P |    |    |    |    |    |    |    |
|---------------------------------------|
5 |    | #P |    |    |    |    |    |    |
|---------------------------------------|
4 |    | ^P |    |    |    |    |    |    |
|---------------------------------------|
3 | ^P |    |    |    |    |    |    |    |
|---------------------------------------|
2 |    |    |    |    |    | ^P |    |    |
|---------------------------------------|
1 |    |    |    |    |    |    |    | ^K |
---------------------------------------
A    B    C    D    E    F    G    H

Diag. 58

The King’s moves are outlined by the necessity of capturing the opposing passed pawn, after which the Black King is two files nearer the battle-field (the Queen’s side), so that the White pawns must fall.

1.  K-Kt2, K-Kt2; 2.  K-Kt3, K-B3; 3.  K-Kt4, K-K4; 4.  P-B4ch, K-B3; 5.  K-Kt3, P-R4; 6.  K-R4, K-B4; 7.  KxP, KxP; 8.  K-Kt6, K-K4, and so on.

For similar reasons the position in Diagram 59 is lost for Black.  White obtains a passed pawn on the opposite wing to that of the King.  He forces the Black King to abandon his King’s side pawns, and these are lost.  I give the moves in full, because this is another important example characteristic of the ever recurring necessity of applying our arithmetical rule.  By simply enumerating the moves necessary for either player to queen his pawn—­separately for White and Black—­we can see the result of our intended manoeuvres, however far ahead we have to extend our calculations.

        1.  P-R4, K-K3; 2.  P-R5, PxP; 3.  PxP, K-Q3

Now the following calculations show that Black is lost.  White needs ten moves in order to queen on the King’s side, namely, five to capture the Black King’s side pawns (K-K4, B5, Kt6, R6, Kt5), one to free the way for his pawn, and four moves with the pawn.  After ten moves, Black only