It has four identical balls (i.e. planets) arranged like a tetrahedron so that they all touch each other, held inside a thin frame. The 'planets' have craters in them, and nestle together so tightly that where two planets touch one or the other must have a crater at that spot. If a planet nestles in craters of at least two of the three others, then it can rotate. There are four coloured regions on each planet, and when the puzzle is solved they are all in the same orientation so that each side of the puzzle has a single colour (blue, red, green, and yellow).
This is rather hard to calculate for two reasons. First of all, it is not entirely clear what constitutes a position, and second, the irregular arrangement of craters blocks many orientations of the planets.
There are only 6 craters on each planet but if we were to add some, extending the arrangement of the existing ones, then a planet would have 12 craters. They are arranged like the vertices of a cuboctahedron. I will only count a planet's orientations where at least two of these 12 points are in contact with the three others planets.
A planet cannot have craters at all three contact points.
There are 8 ways a planet can have craters at two specified contact points (24 ways for craters at any two contacts).
There are 20 ways a planet can have a crater at exactly one specified contact point (60 ways for a single crater at any one contact).
There are 12 ways a planet can have no craters at the three contact points.
Now let's work out the ways the puzzle itself can have craters at the six contact points. Clearly there must be at least one crater at each point, so there are at least 6 craters involved. The planets supply at most 2 of these craters each, so there are very few crater arrangements:
(2,2,2,0): There is only one pattern where three planets have 2 craters, and one planet has none. This pattern can also appear in mirror image.
(2,2,1,1): Again there is only one crater pattern, and its mirror image.
(2,2,2,1): Here there is one symmetric pattern, and two other patterns with their mirror images.
(2,2,2,2): One pattern and its mirror image, as well as another pattern that has mirror and rotational symmetry.
Combining the above two sections, we can find the number of positions:
(2,2,2,0): 12,288 = 2·83
(2,2,1,1): 51,200 = 2·82·202
(2,2,2,1): 51,200 = 5·83·20
(2,2,2,2): 10,272 = 2·84 + (84+82)/2 (using Burnside's lemma for the symmetric crater pattern)
This puzzle is not terribly hard, but there is one observation that makes it a lot easier. The blue and green faces of a planet have only one crater, while the red and yellow ones have two. It is advantageous to have craters inside the puzzle to give you more freedom to move, so it is best to solve the balls with blue and green faces first. Indeed, the solution below does just that.