The *parity argument* is a well known argument telling you
that a Knight takes an *odd number of moves* to go from b1 to
h8, no matter which route is used, because b1 and h8 are squares with
different colors. Similarly, a Knight takes an even number of moves
to go from a1 to h8.

But the parity argument applies to other situations (Knight moves are the simplest case). Consider the following schematic problem:

15+16. Who has the move?

Here each Rook did play an odd number of move. The white Knights did
play an odd total number of moves, and similarly for the black Knights.
The white Queen has been captured at home (by a bN) and the white King
did play an odd number of moves. There remains to account for the 4
moves played by the *a* and *h* Pawns.

Summing all, we conclude that an **odd** total number
of moves has been played, so that **Black must have the move**.
Indeed, the most common use of the parity argument is in
"Who moves?" problems.

Problems using the parity argument are also called to display the "odd/even theme". Here is an other application:

**T. R. Dawson**

The Chess Amateur, Feb. 1927

15+16. Black to play. Indicate a move Black **must**
have played