## Task A-1.5. (4 points)

Eleonora has many cubes, all of whose sides are white. First, she separates one cube and puts it in an empty box. Then she takes one cube at a time and paints some of its sides green, but in such a way that this cube differs from all those already in the box, and then she puts this cube in the box as well. How many cubes can there be in the box at most?
## Solution.

In the box, there is only one cube without any green sides, and one cube with one green side. Also, there is only one cube with all sides painted and one cube with five painted sides.

Cubes with two green sides are not all the same, as the painted sides can be adjacent or opposite. Therefore, there are two such cubes in the box.

Cubes with four painted sides (and two painted sides) are also two.

Finally, there are two cubes with three green sides: on one, all three sides are painted around one vertex, and on the other, two opposite sides and one side between them are painted.

Thus, Eleonora could have put a maximum of $1+1+2+2+2+1+1=10$ different cubes in the box.