A box in the shape of a cube with an edge of $4 \mathrm{~cm}$ is completely filled with arranged dice, small cubes with an edge length of $1 \mathrm{~cm}$. Come up with all different boxes such that they have a square base and all the dice fit exactly into each of them. Write down their dimensions.

(M. Krejčová)
In the described box, there are exactly 64 cubes, because each edge of the box has 4 cubes and $4 \cdot 4 \cdot 4=64$. We are therefore looking for all possible decompositions of the number 64 into a product of three factors, two of which are the same:

- $1 \cdot 1 \cdot 64$,
- $2 \cdot 2 \cdot 16$,
- $4 \cdot 4 \cdot 4$,
- $8 \cdot 8 \cdot 1$.

In addition to the box used in the problem, we can create three more boxes with the following dimensions (the first two figures always correspond to the base): $1 \mathrm{~cm}, 1 \mathrm{~cm}, 64 \mathrm{~cm}$ or $2 \mathrm{~cm}, 2 \mathrm{~cm}, 16 \mathrm{~cm}$ or $8 \mathrm{~cm}, 8 \mathrm{~cm}, 1 \mathrm{~cm}$.

Evaluation. 2 points for calculating the number of cubes in the given box; 1 point for explaining which decompositions of the number 64 need to be found; 1 point for each necessary product and the dimensions of the new box derived from it (i.e., a maximum of 3 points for this part), the product corresponding to the box from the problem and its dimensions should be left without points.

Note. If the solver provides only the information about the dimensions of the box from the problem (i.e., $4 \mathrm{~cm}, 4 \mathrm{~cm}, 4 \mathrm{~cm}$) and no other information that would be scored (e.g., the total number of cubes), do not score this problem with any points. Whether the solver includes or does not include the box from the problem among the sought boxes does not affect the evaluation of the problem.