Five platonic solids. For each one, label all the faces. It does not matter what order or pattern you assign the labels. All that matters is that this assignment is fixed while you rotate the objects.
The Cube is easiest. Pick a face and point it towards some default direction. Now, an axis through the center of this face and the center of the opposite face... Rotate 90 degrees on this axis. You get 4 possible orientations. Pick another face, point it the default way, and rotate it. So we can do this for all 6 faces, so the answer is 24 orientations for the Cube.
On the Tetrahedron, the axis goes through the center of the chosen face and the opposite vertex, but you have to rotate 120 degrees before the faces realign. So there are 4*3 = 12 orientations for the tetrahedron.
I can see the answer for the Dodecahedron too. Rotating 72 degrees will bring the faces back into alignment, so the answer is 12*5 = 60.
...Now we get to the tricky ones, the Octahedron and the Icosahedron. I can't seem to bring the shape back into alignment unless I skip over at least one face. (Btw, I have all these dice and I'm holding them in my hand, rotating them, and I still can't quite wrap my head around this problem for the Octahedron and Icosahedron.)
For the Octahedron, rotating 60 degrees brings up a new face, but the triangle is pointing in the opposite direction. So I'm not sure if the answer is 24 or 48.
For the Icosahedron, very weird but it seems I have to skip over 2 faces before getting a realignment. So I don't know if the answer is 60, 120, or 180.
This doesn't seem right to me. Don't see why I should have to skip over any faces to get a perfect realignment, so I'm wondering if I'm missing something?
Sorry for going on so long but I had to describe my procedure so you knew what "an orientation" means.
In case you're wondering, this came up due to programming 3D dice. Instead of doing a physics engine, it's far easier to just have a preset polygon animation every time you roll...but choose a random orientation at the start. That's why I need to know how many possible orientations there are.