King and Rook

Probability Level 3

Dimitri places a Black King and a White Rook on an empty chessboard. If the probability that Dimitri places the King and the Rook on the chessboard such that the King is NOT in check (that is, the Rook is not attacking the Black King), can be expressed as $\frac{m}{n}$ , in which $m$ and $n$ are coprime positive integers, find $n-m$ .

Details and Assumptions: e.g. If the Rook is on $f2$ , the $f1$ , $f3$ , $f4$ , $f5$ , $f6$ , $f7$ , $f8$ , $g2$ , $h2$ , $e2$ , $d2$ , $c2$ , $b2$ and $a2$ squares are under attack.

The King and the Rook cannot be placed in the same square.

This is the fourth problem of the set Look after the King!

The answer is 2.

2 solutions

Mateo Matijasevick
Apr 11, 2016

Firstly, note that Dimitri can place the Rook in anyone of the $64$ squares. As the King and the Rook can't be placed in the same square, Dimitri has $63$ possible squares to place the Black King. Therefore, there are a total of $64\cdot 63$ possible positions.

Secondly, note that, wherever the Rook is placed, will be attacking its whole row and column. This means that the King can't be placed on the same column or row of the Rook. As each row and column is formed by $8$ squares, and we are counting twice the square on which the Rook is, there are a total of $63-14=49$ squares on which Dimitri can place the King. Therefore, in each of the $64$ possible positions of the Rook, the King can be placed on anyone of the $49$ remaining squares, so the total positions are $49\cdot 64$ .

Thus, the probability is $\frac { 49\cdot 64 }{ 63\cdot 64 } =\frac { 7 }{ 9 }$ . Hence, $m=7$ and $n=9$ , so $n-m=2$ .

Or, (bit simpler):

Given that the King has been placed in any of the squares, there are only 14 places out of the 63 squares left (King is occupying one) that the Rook can be placed and not attack the King.

m ÷ n = 14 ÷ 63 = 7 ÷ 9

n - m = 9 - 7 = 2

Michael Moneke - 5 years, 1 month ago

K T
Nov 15, 2018

After placing the rook, the king can go to 63 other squares, of which 49 are 'safe', no matter where the rook is. He is in check if he is on the same row or file as the rook, which is in 14 out of 63 cases. Now 49÷63=7÷9, and 9-7=2.

King and Rook

This is the fourth problem of the set Look after the King!

The answer is 2.

2 solutions

0 pending reports