Doubly Attacking Queens
Place the maximum number of chess queens on an NxN chessboard, such that
each queen attacks exactly 2 other queens. Solve for N=2,3,4,5.
(I'm interested in higher N, but I think computer searches may be needed
to prove maximality.)
There are two different ways to analyze this:

A queen can attack only as far as another queen. Thus, for three queens in
a line, the center one would be attacked by two queens, but the end queens
would only be attacked by one (the center one.) With this interpretation,
you can easily place N+1 queens on the board by putting N along a side
and 1 in a corner.

A queen can attack through other queens. This simply means that
any line which contains 2 queens can contain no more than 3 queens.
(You could think of the problem then as placing the maximum number of
checkers on the board such that each checker is in line with exactly
two other checkers horizontally, vertically, or diagonally.)
It's my theory that the best solutions will work for both interpretations
above (you won't need to find three queens in a line), but I don't mind being
proven wrong.
Source: Original.
Solution
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