Check Mate!
Two different squares are randomly chosen from an 8x8 chessboard. What is the probability that two queens placed on the two squares can attack each other? Recall that queens in chess can attack any square in a straight line vertically, horizontally, or diagonally from their current position.
Solution:
All squares that are on the edge of the chessboard can hit 21 squares; there are 28 such squares. Now consider the 6x6 chessboard that is obtained by removing these bordering squares. The squares on the edge of this board can hit 23 squares; there are 20 of these squares. Now we consider the 12 squares on the boundary of the 4x4 chessboard left; each of these squares can hit 25 squares. The remaining 2 can hit 27 squares. The probability then follows as:
(21 × 28 + 23 × 20 + 25 × 12 + 27 × 4) / (64 × 63) =13/36
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Solution:
All squares that are on the edge of the chessboard can hit 21 squares; there are 28 such squares. Now consider the 6x6 chessboard that is obtained by removing these bordering squares. The squares on the edge of this board can hit 23 squares; there are 20 of these squares. Now we consider the 12 squares on the boundary of the 4x4 chessboard left; each of these squares can hit 25 squares. The remaining 2 can hit 27 squares. The probability then follows as:
(21 × 28 + 23 × 20 + 25 × 12 + 27 × 4) / (64 × 63) =13/36
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