With Bishop, Knight and King versus King, the Checkmate is attainable, but in a complicated manner. The principle of the hunt for the King is this: the King, the Bishop and the Knight in their effort to chase the adverse King into a corner, must apportion different tasks to each other, thus aiding each other; they must not perform any task doubly which would amount to an unnecessary competition between each other and would be a useless effort. Each one of these pieces must therefore cut off squares to the hunted King that are not cut off by the other pieces, each one must have its own allotted share of squares to guard and must not trespass. With this principle in view, it is not hard to drive the King into a corner. But now another difficulty arises. The adverse King must be driven into one of those corners that the Bishop can assail, as mate cannot be forced in either of the other corners. If the Bishop moves on white squares, it is limited to these white squares, for diagonals running through a white square comprise white squares only; and with black squares it is the same. Hence, the hunted King will be prudent to allow itself to be driven towards a corner unattainable to the Bishop and to shun either corner of the colour assailable by the Bishop. On the other hand, the assailant, having driven the King into the safe corner, is confronted by the problem of driving it into one of the unsafe corners.
This latter task has to be studied in detail. The play runs as follows:
1 K-Kt6
K-Kt1
2 K-B6
K-R1
3 Kt-Q5
K-Kt1
4 Kt-B7
Thus the corner is cut off.
4 ... K-B1
5 B-R7
K-Q1
6 Kt-Q5
K-K1
Black makes an attempt to gain more freedom of movement.
7 K-Q6
K-B2
8 Kt-K7
K-B3
9 B-K3
The net holds tight, the attempt of the Black King is frustrated.
9 ... K-B2
10 B-Q4
K-K1
11 K-K6
K-Q1
12 B-Kt6ch.
K-K1
Now the Knight has to cut off the square K8.
13 Kt-B5
K-B1
14 Kt-Q6
K-Kt2
15 K-B5
K-R3
16 B-Q8
K-R4
17 Kt-K8
K-R3
18 B-Kt5ch.
The Black King dares not return, else it is Checkmated at once.
18 ... K-R2
19 K-B6
K-Kt1
20 K-K7
K-R2
21 K-B7
K-R1
22 B-R6
K-R2
23 B-B8
K-R1
24 B-Kt7ch.
K-R2
25 Kt-B6Mate.