# 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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