A rook and a bishop of a standard chess set and having different colors are randomly placed on a standard chessboard.

Determine the probability that one is attacking the other.

The problem is equivalent to two mutually attacking queens of different color.  So for each space the first queen is on we must find the number of squares it attacks.

A queen on any of the 28 the outer edge spaces (including corners) attacks 7 ranks, 7 files and 7 diagonals for a total of  21.
21*28 = 588
A queen on any of the 20 squares that are one square from the edge can attack an additional 2 diagonals.
23*20 = 460
A queen on any of the 12 squares that are two squares from the edge can attack two more diagonal squares.
25*12 = 300
Finally a queen on one of the 4 central squares can attack two more diagonals squares
27*4 = 108

For a total of 1456 attacking combinations.

There are 63*64=4032 ways of placing two queens of different color.

1456/4032 = 13/36 is the requested probability.

Posted by Jer on 2014-05-14 13:19:31