In how many ways can two queens be placed on a chessboard such that they are mutually attacking each other?

Show how to solve this problem quickly with only pencil and paper (not even a calculator).

Consider the 4 squares like a1, or the 8 squares like a2, or the 8 like a3, or the 8 like a4: the other queen may be in 21 places --> 28x21=588.

For the 4 squares like b2, or the 8 like b3, or the 8 like b4, the other queen may be in 23 places --> 20x23=460.

For the 4 squares like c3, or the 8 like c4, the other queen may be in 25 places --> 12x25=300.

For the 4 squares like d4, the other queen may be in 27 places --> 4x27=108.

The total number of possibilities, if the queens were of different color, would be 1456; if they are the same color, halve this to get 728.