Proofs of Existence and Optimality · Proofs of Nonexistence
Lesson 9
Let's provide a mathematically rigorous proof.
- For the queens, this is the simplest: in each column, there can be no more than one queen, so there will definitely be no more than eight queens.
- To show that there cannot be more than 14 bishops, one can consider 15 diagonals in one direction (covering the entire board) and notice that it is not possible to place bishops on the first and last of them simultaneously.
- To show that more than 32 knights cannot be placed, one can divide all the squares of the board into pairs (see the figure), so that in each pair no more than one square can be used.
