Dwarves painted cubic dice with green and white colors so that each face was entirely painted in one of these two colors. After a while, they noticed that some of the painted dice looked exactly the same after a suitable rotation and began to sort them into groups (in the same group are identically painted dice).

What is the maximum number of groups they could thus obtain?

(I. Jančigová)

Hint. What relationships can pairs of cube faces have?
A cube has six faces, with each face adjacent to four other faces (they share an edge) and parallel to one face (they share no points).

We can categorize the colorings based on the number of green (or white) faces. This gives us seven possibilities, for each of which we will analyze different types of coloring.

- No green faces (all white): all such cubes look the same, so we have only one type.

![](https://cdn.mathpix.com/cropped/2024_04_17_cc913e50e7898f0d1267g-2.jpg?height=252&width=280&top_left_y=1733&top_left_x=891)

- One green face (five white): likewise, only one type.

![](https://cdn.mathpix.com/cropped/2024_04_17_cc913e50e7898f0d1267g-2.jpg?height=260&width=283&top_left_y=2143&top_left_x=892)

- Two green faces (four white): we distinguish two types based on whether the green faces are adjacent or not.
![](https://cdn.mathpix.com/cropped/2024_04_17_cc913e50e7898f0d1267g-3.jpg?height=258&width=642&top_left_y=196&top_left_x=708)
- Three green faces (three white): we distinguish two types, depending on whether the green faces are adjacent in pairs or all mutually adjacent.
![](https://cdn.mathpix.com/cropped/2024_04_17_cc913e50e7898f0d1267g-3.jpg?height=268&width=644&top_left_y=711&top_left_x=708)
- Other cases do not need to be listed: the discussion for options with swapped numbers of green and white faces is the same.

In total, we get $1+1+2+2+2+1+1=10$ types of coloring. The dwarves could receive a maximum of 10 groups of colored cubes.

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