Solution
Two Peaceful Queens
Show the problem again
Two identical queens are placed on two different squares of a standard 8 × 8 chessboard. In how many ways can this be done so that the queens do not attack each other (that is, they share no row, no column, and no diagonal)? Give an integer.
Worked solution
The answer is 1288. There are C(64, 2) = 2016 unordered pairs of squares. Count the attacking pairs. Same row: 8 rows × C(8, 2) = 224. Same column: also 224. Diagonals in one direction have lengths 1, 2, ..., 7, 8, 7, ..., 1, giving 2(C(2,2) + C(3,2) + ... + C(7,2)) + C(8,2) = 2 · 56 + 28 = 140 pairs, and the other direction gives another 140. Two squares share at most one of these lines, so there are 224 + 224 + 280 = 728 attacking pairs and 2016 − 728 = 1288 peaceful ones.
Source: Classic counting exercise in wide circulation; no traceable original source. Statement written for AxiomIQ.