# A brain teaser to shake off winter doldrums?

If you take a square, there are four ways that you can rotate it so that it occupies the same space. How many ways can you rotate a cube so it occupies the same space?

If you look at it the right way the solution will be immediate. I like this one because it forces you to look at something familiar in a different way. There is also another interesting aspect involving rotation axes that I will discuss when giving the solution.

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## 10 Answers

24? I just wanna know the answer now because I am sure it is something obvious I am overlooking.

I’m thinking six? so what’s the catch… It’s obviously something in the rotation and how many times you rotate it? or is it directional? or am I just simple…

Oh c’mon that’s not nice…

It’s tempting to simply multiply the number of possible turns of a square by the number of faces on a cube. In this case, 4×6 = 24. However, in 3 dimensional space, the turns of one face will be the same as the opposite rotation of the opposite face. In other words, you can’t count the faces that are parallel to another face.

Using that logic, I believe the correct answer is 12.

What do I win?

Those of you who said 24 got it right. There are 6 faces that can be on the bottom and for each you can rotate the bottom face four times.

Now here is the connection to rotation axes. If you fix the center of any object in space then it may or may not be intuitively obvious to you that for any orientation, you can reach it by a single rotation about an appropriate axis through the center.

So what are the rotation axes for a cube. Everyone knows the three obvious ones, the lines through the centers of opposite faces. How many orientations do these axes account for. Each one allows for 4 positions, but the starting position is the same for each, so each axis only contribute 3 positions in addition to the starting one. The total is therefore 1 + 3 + 3 + 3 = 10, far short of the total of 24.

If, like me, you find it hard to visualize in 3 dimensions, this site shows the other rotation axes:

http://www.luc.edu/faculty/spavko1/minerals/prelims/plato/cube-main.htm. The site designates the above axes as C4.

There are two other types of axis. One goes through the long diagonal and are designated as C3 at the Web site. There are 4 of them. At the end of the diagonals, 3 faces come together, so each of these axes has 3 postions and contribute 4*2 =8 additional positions.

The other rotation axis goes through the centers of two “diagonally opposite” sides and are designated C2. The 12 sides of the cube can be divided up into 6 pairs of such sides and each axis has 2 positions, accounting for 6*1=6 new positions.

Adding the total for each type of axis gives 10 + 8 + 6 = 24 postions.

I say 4. You can rotate it left, right, forward, back.

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