Long ago, I invented a problem which solution looks like this:
The problem was to place a complete set of white chess pieces on the chessboard such that the number of bonds between them is maximal. Here you see that the king protects 8 neighboring pieces and the pawn in the top line covers no piece. The number of all protections is 53 and it can’t be done any better. (You may notice a pawn in the first line and those bishops are of the same color, but that’s fine.)
If you now add 40 more queens and rearrange all those 56 pieces, then there will be at most 338 bonds in total. We call that number B – the maximal bonding number for the set F. For all the white pieces it’s 53, for 2 knights only it’s 2, and so on.
We have a set F with bonding number B1. And a proper subset F’ with the bonding number B2. B1 is equal to B2. Which set is that?
In other words, for which set you can remove one or more pieces and the maximal binding number remains the same as it was initially.
There is a trivial solution of any single piece and the empty set, but as always, we want a nontrivial solution.