Thanks to @Gamow's comment, this number's maximality can be proved
by self-contradiction of the assumption that it is not maximal.
Any more dominos would cover all 64 squares.
Suppose that we have a chessboard with the desired properties.
Find the greatest number in each row. Out of these numbers, let the smallest be $m_i$ in row $i$.
Find the smallest number in each row....
I have a computer program for solving packing problems, and found a way to use it to solve this problem. One of the solutions it found is below:
Note that this is very close to the attempted solution ...
Here's the solution:
There's a very neat method for finding this, inspired by the no-computers way of solving another related puzzle. Namely,
More specifically, given the constraints of this problem:...
50 Kings,14 Knights:
This is optimal but not unique, see bottom of this answer.
I think the problem is equivalent to covering every square on the board with as few knights as possible and ...