King and Pawn

Discrete Mathematics Level 3

Dimitri places a Black King and a White Pawn on an empty chessboard. He cannot place the Pawn on the first or eighth row (\(1\) or \(8\)). If the probability that Dimitri places the King and the Pawn on the chessboard such that the King is \(NOT\) in check (that is, the Pawn is not attacking the Black King), can be expressed as \(\dfrac{m}{n}\), in which \(m\) and \(n\) are coprime positive integers, find \(n-m\).

As an explicit example, if the Pawn is on \(f2\), the \(g3\) and \(e3\) squares are under attack, then the White Pawn attacks only the \(2\) (at most) squares immediately forwardly and diagonally.

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

This problem is the first one of the set Look after the King!

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