# A Bishop And A Rook Puzzle

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A white rook and a black bishop of a standard chess set are randomly placed on a chessboard

What is the probability that one is attacking the other?

Detective Expert Asked on 15th October 2015 in

EXPLANATION-
The probability of either the Rook or the Bishop attacking the other is 0.3611

A Rook and a Bishop on a standard chess board can be arranged in:
= 64P2  (64 Permutation 2)
= 64*63
= 4032 ways

Now,
There are 2 possible cases – Rook attacking Bishop and Bishop attacking Rook.

As we know that the Rook and the Bishop never attack each other simultaneously. Let’s consider both the cases one by one.

Case I – Rook attacking Bishop
Rook can be placed in any of the given 64 positions and it always attacks 14 positions.
Hence, total possible ways of the Rook attacking the Bishop
= 64*14
896 ways

Case II – Bishop attacking Rook
View the chess-board as a 4 co-centric hollow squares with the outermost square with side 8 units and the innermost square with side 2 units.

If the bishop is in one of the outer 28 squares, then it can attack 7 positions.
If the bishop is in one of the 20 squares at next inner-level, then it can attack 9 positions.
If the bishop is in one of the 12 squares at next inner-level, then it can attack 11 positions.
If the bishop is in one of the 4 squares at next inner-level (the innermost level), then it can attack 13 positions.

Hence, total possible ways of the Bishop attacking the Rook
= 28*7 + 20*9 + 12*11 + 4*13
= 560 ways
Thus, the required probability is
= (896 + 560) / 4032
= 13/36
= 0.3611

SherlockHolmes Expert Answered on 16th October 2015.

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