King and King

Discrete Mathematics Level 4

Dimitri places a Black King and a White King on an empty chessboard. If the probability that Dimitri places both Kings on the chessboard such that neither of the Kings is in check (that is, the White King is not adjacent to 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 a King is on \(\text f2\), the \(\text f1\), \(\text g1\), \(\text e1\), \(\text f3\), \(\text g2\), \(\text e2\), \(\text e3\) and \(\text g3\) squares are under attack and these are the adjacent squares to \(\text f2\). Note that if the other King is placed on any of the squares, the Kings are attacking each other, so both are in check, which is impossible.

The White King and the Black King cannot be placed in the same square.

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

Problem Loading...

Note Loading...

Set Loading...