Chess is more than counting wood!

As a beginner we learn the different values of pieces: horsey and bishop are both roughly worth 3 pawns, a rook is worth 5 pawns and so forth. In the following examples the material balance is clearly in favor of Black (rook and pawn against bishop) and it is Black to move. Please evaluate the three positions one by one:

Let's compare the three positions:

Please look at the position below! It is balanced, but the evaluation is not trivial at all! Black has to do something in order to prevent checkmate Bxf7# after an invasion of the white king on f6 or e7. Black has to somehow activate the rook and liberate the king. What can Black do?

1. A knight on e6 would unpin f7, but that does not help: the pawn on f7 has to protect the knight then.
2. Push away the bishop from h6. The only way to do that is to bring the knight to f5, but White can play g4 to prevent that.
3. Bring the knight to g7 in order to enable Kf8

The third plan is doable, but White has enough time to grab e5 and the activity of the bishops ensures a draw:

With queens Black is worse:

So in the first position the material advantage of Black is decisive, in the second position Black can only hold a draw due to the white piece activity. In the last position due to the presence of queens the rook cannot be activated, therefore the value of the rook is close to zero.

Take-away

The value of a piece depends on the activity. For instance an active bishop or horsey can compensate or even overcompensate a passive rook.

Test

The basic inspiration to create this blog post came from this forum post: https://lichess.org/forum/lichess-feedback/why-lichess-choose-puzzles-from-rapid-and-classical-game-only#6 . The test here is one move after the final position of the mentioned puzzle, where Black has to play Kc7 after Nxf7+: https://lichess.org/training/jucpS .