Yes, it's another "Chess Pieces Attacking Each Other" puzzle. This time we have 3 colors. Your goal is to place as many of the same type of chess piece (excluding pawns since you can't define the "forward" direction for 3 players) as possible onto an 8x8 board in 3 colors such that each piece attacks exactly 1 piece from each of the other colors. Pieces may attack any number of their own color, these are ignored. Note that there will necessarily be the same number of each color in every solution.
I have come up with a number of potentially optimal solutions (by hand, not verified by computer). The number of pieces for each is hidden in the following hint. Don't look if you'd rather see how many you can get on your own without knowing where the lower bar is, though I suspect that there is room for improvement on some of these.
Bishops: 14 each
Knights: 10 each
Rooks: 20 each (fairly certain this is optimal)
Queens: 6 each
Kings: 12 each