Don't Think Outside The Box!

Geometry Level 2

Brilli the ant is trapped on a cube-shaped planet. She wants to go from one corner [point\(A\)] to the opposite corner [point \(B\)]. However she wants do this in a way such that she has to cross the shortest distance possible. If the length of the sides of the cube is \(1\), the shortest distance between \(A\) and \(B\) can be expressed as \(p+\sqrt{q}\) where \(p\) and \(q\) are non-negative integers and \(q\) is square-free. What is \(q-p\)?

Details and assumptions:

Brilli the ant is completely confined to the surface of the cube. She can't move inside or outside the cube.

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