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.