Chess/Puzzles/Placement/14 Bishops/Solution
There are several solutions to this puzzle, but they are all quite similar.
Here's a possible one:

Proof of maximalityEdit
There are 15 diagonals on the chessboard running from bottom left to top right. They are:
 a8a8
 a7b8
 a6c8
 a5d8
 a4e8
 a3f8
 a2g8
 a1h8
 b1h7
 c1h6
 d1h5
 e1h4
 f1h3
 g1h2
 h1h1
Each of these diagonals can only contain one bishop. Also, the first and last diagonals cannot both contain a bishop, since both are on the diagonal a8h1. Therefore, we can place at most 13 bishops on the other 13 diagonals, and one bishop on those two diagonals, for a total of 14 bishops. Since 14 bishops is possible, 14 is the maximum number of bishops we can place so no two attack each other.