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Chessboard (Posted on 2004-01-14) Difficulty: 3 of 5
Three different squares are chosen randomly on a chessboard.

What is the probability that they lie in the same diagonal?

No Solution Yet Submitted by Ravi Raja    
Rating: 3.5000 (8 votes)

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Solution Corrected solution | Comment 5 of 20 |
(My original entry had to be corrected after the criticism by SilverKnight. I had overlooked the fact that the first square could be on the same diagonal as the 2nd square and the 3rd square, but the 2nd and 3rd squares might not be on the same diagonal with respect to each other).

The correct answer: the probability that three randomly chosen squares on a chessboard are all on the same diagonal is less than 1%

Explanation:

Each corner square on the chessboard in on just one diagonal. Every other square is on two diagonals. I overlooked this in my original solution.

If we adopt the modern algebraic notation for the chessboard, where the lower left-hand square is a1 and the upper right-hand corner is h8, then, of the 64 squares on the chessboard:

a1, h1, a8 and h8 are each on one diagonal with 7 other squares.
b1, g1, a2, h2, a7, h7, b8, and g8 are on diagonals with 1 and 6 other squares.
c1, f1, a3, h3, a6, h6, c8, and f8 are on diagonals with 2 and 5 other squares.
d1, e1, a4, h4, a5, h5, d8, and e8 are on diagonals with 3 and 4 other squares.
b2, g2, b7, and g7 are on diagonals with 2 and 7 other squares.
c2, f2, b3, g3, b6, g6, c7, and f7 are on diagonals with 3 and 6 other squares.
d2, e2, b4, g4, b5, g5, d7, and e7 are on diagonals with 4 and 5 other squares.
c3, f3, c6, and f6 are on diagonals with 4 and 7 other squares.
d3, e3, c4, f4, c5, f5, d6, and e6 are on diagonals with 5 and 6 other squares.
e4, f4, e5, and f5 are on diagonals with 6 and 7 other squares.


So the true odds are:

(4/64)*(7/63)*(6/62)
+ (8/64)*(6/63)*(5/62)
+ (8/64)*(2/63)*(1/62) + (8/64)*(5/63)*(4/62)
+ (8/64)*(3/63)*(2/62) + (8/64)*(4/63)*(3/62)
+ (4/64)*(2/63)*(1/62) + (4/64)*(7/63)*(6/62)
+ (8/64)*(3/63)*(2/62) + (8/64)*(6/63)*(5/62)
+ (8/64)*(4/63)*(3/62) + (8/64)*(5/63)*(4/62)
+ (4/64)*(4/63)*(3/62) + (4/64)*(7/63)*(6/62)
+ (8/64)*(5/63)*(4/62) + (8/64)*(6/63)*(5/62)
+ (4/64)*(6/63)*(5/62) + (4/64)*(7/63)*(6/62)

0.0094086022

Edited on January 14, 2004, 7:44 pm
  Posted by Penny on 2004-01-14 19:00:46
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