What is the most number of black and white queens that you can place on a standard 8x8 chess board, such that each queen attacks exactly one opponent queen?
Following @Konstantin Pfennig's answer, using black and white queens, and squeezing in more, I have:
You can only place at most
If each queen attacks exactly one other queen, then the queen being attacked is also attacking the original queen. So the queens have to come in pairs. Now, the board only has 8 rows and 8 columns. For each pair of queens, they will necessarily take either 2 rows and 1 column, 2 columns and 1 row, or 2 rows and 2 columns if they are attacking each other on the diagonal.
You have probably guessed that having them attack on the diagonal is not optimal. With these constraints, you can place at most 5 pairs. Any more and you will need either more than 8 rows or more than 8 columns.
Here is one solution:
It is possible to have higher queen density with blocking pieces.
I managed 19 pairs, 38 queens using pretty simple 3x3 tile that achieves highest density without blockers - but blockers must be added to replicate the pattern.
This approach uses the highest density 3x3 tile seen in top-left corner (for 3x3 there is no need for blocking pieces). The last two rows and columns are simple 1/2 density that is simple enough to achieve with blockers.
It may be possible to do even better, I am not too satisfied with the last 2 rows and columns of the solution above.
2xN tiles are obvious to be limited to mere 1/2 density with matching queens, so don't bother wiggling those queens on last two rows/columns. I also haven't found a good 4x4 tile with higher density and I suspect it does not exist. But it may be possible to use 5x5, 3x5 and 3x3 tiles to achieve high resulting density and place 20 pairs on the chessboard. But checking all reasonable options on the 5x5 is a huge pain.
I believe it's 8, and here is a possible solution. Every pair of attacking queens must exclusively occupy at least 1 row and 2 columns or 2 rows and 1 column, making any number greater than 8 impossible because aren't enough rows or columns to contain >4 pairs.