Question

What is the maximum number of coins that can be placed on squares in an 8x8 chessboard such that each row, each column, and each long diagonal contains at most 4 coins? (Note: Only 1 coin is allowed per square.)

Answer 1

If long diagonal is defined as continuous diagonals with 8 squares, then there are only two.
The following scheme should work.
C C _ _ C C _ _
_ _ C C _ _ C C
C C _ _ C C _ _
_ _ C C _ _ C C
C C _ _ C C _ _
_ _ C C _ _ C C
C C _ _ C C _ _
_ _ C C _ _ C C

Each row and each column contains exactly 4 coins, and each of the two long diagonal also contains 4 coins.
In fact, each discontinuous diagonal also contains four coins.
So the total number of coins is 4*8=32.