Simple end-games, that is, end-games without pawns, are comparatively easy to understand.  Let us first consider the case of a King denuded of all his troops.  In order to force the mate it is necessary to obtain command of four squares, namely, those four squares which he controls after he has been driven into a corner.  Supposing the Black King has been driven to QR1, the White King can prevent him from reaching two squares of different colour, namely, QR2 and QKt2.  Therefore it is necessary for White still to have such forces as can command two more squares of different colour, namely, QR1 and QKt1.  As can readily be seen, it will be essential to have at least the Queen or a Rook or two Bishops, or a Knight and Bishop, or two Knights. [Footnote:  How the King can be driven into a corner will be shown subsequently.]

We shall see that in the latter case it is impossible to drive the King into a corner without bringing about a stalemate.  The mates by a Queen or Rook are so simple that I only give an example of each for the sake of completeness.

Position 1.—­White:  K at QR1, Q-KR1
Black:  K at K4

1.  K-Kt2, K-Q5; 2.  K-Kt3, K-K4; 3.  K-B4, K-Q3; 4.  Q-K4, K-Q2; 5.  K-B5, K-B1; 6.  K-B6, K-Kt1; 7.  Q-QR4, or Kt4ch, or K7, or R7 and mate next move.

Position 2.—­White:  K at QKt3, RKR2
Black:  K at K4

1.  K-B4, K-Q3; 2.  R-K2, K-B3; 3.  R-K6ch, K-Q2; 4.  K-Q5, K-B2; 5.  K-B5, K-Q2; 6.  R-K1, K-B2; 7.  R-K7ch, K-Q1; 8.  K-Q6, K-B1; 9.  K-B6, K-Kt1; 10.  R-K1, K-R7; 11.  R-K8, K-R3; 12.  R-R8 mate.

Position 3.—­White:  K at QRsq, B at KKtsq, BatKKt2
Black:  K at KRsq

1.  K-Kt2, K-Kt2; 2.  K-B3, K-B3; 3.  K-Q4, K-K3; 4.  B-R2, K-B3; 5.  K-Q5, K-B4; 6.  B-K5, K-Kt4; 7.  K-K6, K-Kt5; 8.  B-QR8, K-Kt4; 9.  B-B3, K-Kt3; 10.  B-KB6, K-R3; 11.  K-B7, K-R2; 12.  B-Kt5, K-R1; 13.  B-Q1, K-R2; 14.  B-B2ch, K-R1; B-B6 mate.

It is more difficult to mate with knight and bishop.  It is only possible to mate on a corner square commanded by the Bishop, as the following argument shows clearly.  A mating position in the corner which the Bishop does not command would have to be of the type set out in Diagram 42.  Here the Bishop plays on White squares, and the Knight in order to checkmate must move on to a White square; in other words, he must come from a Black one.  Therefore, when the Bishop checked on the previous move and drove the King away, the King had the option of two black squares, and had no need to go into the corner one.  He is only mated in consequence of a wrong move.

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8 |    |    |    |    |    | ^K |    | #K |
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7 |    |    |    |    |    | ^Kt|    |    |
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6 |    |    |    |    |    |    |    |    |
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5 |    |    |    |    |    |    |    |    |
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