Is it possible to position all chess pieces on the board such that all pieces defend at least one piece and attack at least one piece?
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Yes, and it's not particularly hard to do. Here's one solution.
I found this by placing the pawns first (because they are much more limited in their attack/defence potential), and then adding the pieces one at a time. Using this simple manual approach, it took about 15 minutes to find a solution. As an added constraint, I avoided placing the kings in check.
There are undoubtedly many alternative arrangements that would also work. It would be quite straightforward to write a computer program to search for valid configurations. A simple brute-force backtracking algorithm would probably suffice.