King and Bishop

Probability Level 4

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

As an explicit example, if the Bishop is on $f2$ , the $g1$ , $e1$ , $g3$ , $h4$ , $e3$ , $d4$ , $c5$ , $b6$ and $a7$ squares are under attack.

Dimitri can place the Bishop either on a white or black square.

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

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

The answer is 67.

1 solution

Sam Bealing
Apr 17, 2016

We begin by placing the king in one of four concentric rings of squares of the chessboard. Each of these rings has a fixed number of squares that the bishop can be placed on the board:

Ring	Squares in ring	Squares to place bishop	Probability
$1$	$8^2-6^2=28$	$63-7=56$	$\frac{56 \times 28}{64 \times 63}$
$2$	$6^2-4^2=20$	$63-9=54$	$\frac{54 \times 20}{64 \times 63}$
$3$	$4^2-2^2=12$	$63-11=52$	$\frac{52 \times 12}{64 \times 63}$
$4$	$2^2-0^2=4$	$63-13=50$	$\frac{50 \times 4}{64 \times 63}$

This gives our probability as:

$\frac{1568}{4032}+\frac{1080}{4032}+\frac{624}{4032}+\frac{200}{4032}=\frac{31}{36}$

$36+31=\boxed{67}$

Moderator note:

Good simple approach using the rule of sum.

King and Bishop

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

The answer is 67.

1 solution

Moderator note:

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