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I'll give the general solution for a nxnxn cube.
When painted 6n^2 unit squares will be painted. There are 6n^3 total sides. The chosing and rolling assures that each of these total sides has the same probability of being chosen.
(6n^2)/(6n^3)=1/n.
So the probability when n=7 is 1/7
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Another very slick way of looking at this.
Consider the top face of the big cube. Each little cube has paint on this side, but there are n-1 cubes below it that don't. The same holds in each direction so the answer is 1/n.
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