exchanges on one square, all you have to do is to count the
number of attacking and defending units, and to compare their
relative values; the latter must never be forgotten. If Black
were to play KtxP in the following position, because the pawn at
K 5 is attacked three times, and only supported twice, it would
be an obvious miscalculation, for the value of the defending
pieces is smaller. [Footnote: It is difficult to compare the
relative value of the different pieces, as so much depends on the
peculiarities of each position, but, generally speaking, minor
pieces, Bishop and Knight, are reckoned as equal; the Rook as
equal to a minor piece and one or two pawns (to have a Rook
against a minor piece, is to be the "exchange" ahead). The Queen
is equal to two Rooks or three minor pieces.]

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

Diag. 5.

Chess would be an easy game if all combinations could be tested
and probed exhaustively by the mathematical process just shown.
But we shall find that the complications met with are extremely
varied. To give the beginner an idea of this, I will mention a
few of the more frequent examples. It will be seen that the
calculation may be, and very frequently

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

Diag. 6.

is, upset by one of the pieces involved being exchanged or
sacrificed. An example of this is found in Diagram 6; KtxP

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

Diag. 7.

fails on account of R X B; this leaves the Knight unprotected,
and White wins two pieces for his Rook. Neither can the Bishop
capture on K5 because of R X Kt. leaving the Bishop unprotected,
after which BxKt does not retrieve the situation because the Rook
recaptures from B6.

A second important case, in which our simple calculation is of no
avail, occurs in a position where one of the defending pieces is
forced away by a threat, the evasion of which is more important
than the capture of the unit it defends. In Diagram 7, for
instance, Black may not play KtxP, because White, by playing P-
Q6, would force the Bishop to Kt4 or B1, to prevent the pawn from
Queening and the Knight would be lost. A further example of the
same type is given in Diagram 8. Here a peculiar mating threat,
which occurs not

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

Diag. 8.

infrequently in practical play, keeps the Black Queen tied to her
KB2 and unavailable for the protection of the B at BI.

White wins as follows:

1. KtxB, KtxKt; 2. RxKt, QxR; 3. Kt-B7ch, K-Kt1; 4. Kt-R6 double
ch, K-R1; 5 Q-Kt8ch, RxQ; 6. Kt-B7 mate.

We will now go a step further and turn from "acute" combinations
to such combinations as are, as it were, impending. Here, too,
I urgently recommend beginners (advanced players do it as a
matter of course) to proceed by way of simple arithmetical
calculations, but, instead of enumerating the attacking and
defending pieces, to count the number of possibilities of attack
and defence.

Let us consider a few typical examples. In Diagram 9, if Black
plays P-Q5, he must first have probed the position in the
following way. The pawn at Q5 is attacked once and supported once
to start with, and can be attacked by three more White units in
three more moves (1. R-Q1, 2. R(B2)-Q2, 3. B-B2) Black can also
mobilise three more units for the defence in the same number of
moves (1. Kt-B4 or K3, 2. B-Kt2, 3. R-Q1). There is,
consequently, no immediate danger, nor is there anything to fear
for some time to come, as White has no other piece which could
attack the pawn for the fifth time.

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

Diag. 9.

It would be obviously wrong to move the pawn to Q6 after White's
R-Q1, because White could bring another two pieces to bear on the
P, the other Rook and the Knight, whilst Black has only one more
piece available for the defence, namely, his Rook.

The following examples show typical positions, in which simple
calculation is complicated by side issues.

In Diagram 10, the point of attack, namely, the Black Knight at
KB3, can be supported by as many Black units as White can bring
up for the attack, but the defensive efficiency of one of Black's
pieces is illusory, because it can be taken by a White piece. The
plan would be as follows: White threatens Black's Knight for the
third time with Kt-K4, and Black must reply QKt-Q2, because
covering with R-K3 would cost the "exchange," as will appear from
a comparison of the value of the pieces concerned. The "exchange"
is, however, lost for Black on the next move, because

---------------------------------------
8 | #R | #Kt| #B | #Q | #R | | #K | |
|---------------------------------------|
7 | | #P | #P | | | #P | #B | #P |
|---------------------------------------|
6 | #P | | | #P | | #Kt| #P | |
|---------------------------------------|
5 | | | | ^Kt| #P | | ^B | |
|---------------------------------------|
4 | | | | ^P | | | | |