Consider a cube C1 (k=3 dimension) and suppose it has the length of a side n=3 (as the (g)old

Rubik's cube).

- How many cubes there are on the surface of C1?

Now consider a cube C2 (k=3 dimension) and suppose it has the length of a side n (see

Rubik Variations in wikipedia for examples of n=4, 5, 6, etc)

- How many cubes there are on the surface of C2?

Now consider an

hypercube, a generalization of a cube in k dimensions.

- Let's start with k=4 dimensions. How many hypercubes there are on the surface of n * n * n * n hypercube with k=4?
- Now, generalize to k generic dimensions. How many hypercubes there are on the surface of n * n ..... n * n (k times) hypercube with k dimensions?

I like this problem since it's hard to guess a formula for the final question, but it is easy to derive it if you solve simpler problems first.

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