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What is the largest positive integer that can't be made by adding up 5's and 7's?

You and a friend play a game where you take turns painting the little squares on the sides of a \(3 \times 3 \times 3\) Rubik's cube.

Each time ...

Suppose you have a \(2 \times 2 \times 2\) cube and you can color each of the 24 little squares one of ...

A bug starts on a vertex of a cube.

If he only moves along the edges, how many ways can he travel to the opposite vertex visiting each vertex along ...

This is a dodecahedron:

Let's suppose all sides are painted either \(\color{Red} \text{red}\), \(\color{Green} \text{green}\) or \(\color{Blue} \text{blue}\).

And at one vertex they ...

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