Thomas sent in the solution below. We also received a correct solution from Andrei of School 205 Bucharest as well as a number of partial solutions.

The description of the thirteen axes of revolution are as follows:

three of them penetrate the cube through the centre of a given face and exit through the center of the opposite face (using a die as an example, through the 1 and 6, 2 and 5, and 3 and

four of them penetrate the cube through a given vertex and exit through the opposite vertex (if the cube were to be stood on a vertex, it would exit through the one on top);

the other six enter the cube through the midpoint of a given edge and exit through the midpoint of the opposite edge (if the cube were stood on edge, it would exit through the midpoint of the top edge).

As for the mean length of the axes, the three axes through the face have the same length as an edge (call that length

s

Cube with axis of rotational symmetry joining oppostie faces

the six edge axes have a length equal to the diagonal of a face, or

s \times \sqrt 2

and the four vertex axes, the hypotenuses of right triangles with sides of

s

and

s \times \sqrt 2

, have length:

s \times \sqrt 3

Summed up, the average axis length is:

\frac{(3 + (6 \times \sqrt 2) + (4 \times \sqrt 3) \times s}{13}