Counting the number of rotations of a cube. 
Take a cube $C$ $[-1,1]^3\subset\Bbb{R^3}$. How many rotations are there which take $C$ to itself?

What does this question even mean? If you take any line in $\Bbb{R^3}$, and rotate the cube around it by $2\pi$ rad, you are mapping the cube to itself!
Ans: 24. I don't understand.
 A: Let $A = \{ \pm e_k \}$, and suppose $Q$ is a rotation that satisfies $QC \subset C$. Then we will show that $QA = A$, which will restrict the possibilities for $Q$.
Since $Q$ is a rotation, we have $\|Qx\|= \|x\|$, and so $Q \partial B(0,1) = \partial B(0,1)$. A rotation is invertible and the inverse is also a rotation, so we
have $Q B(0,r) = B(0,r)$ for all $r >0$.
We must have $QC = C$. There are various ways of seeing this. Suppose the containment is strict, then $C$ contains some open ball that does not
intersect $QC$, which means that the volumes satisfy $m(QC) < m(C)$, which
contradicts $Q$ being a rotation. We also have $Q(\lambda C) = \lambda C$ for
all $\lambda$ (where $\lambda C = \{ \lambda c \}_{c \in C}$).
We also have $C^\circ = \cup_n (1-{1 \over n}) C$, and so it follows that
 $Q C^\circ = C^\circ$.
Since $C$ is given by the disjoint union $C = \partial C \cup C^\circ$,
we have $Q \partial C = \partial C$.
Note that $A=\partial C \cap \partial B(0,1)$, 
combining the above shows that $QA = A$.
Now count the possibilities for $Q$: 
Since $e_1,e_2,e_3$ form a basis, we need only consider the action of $Q$ on these vectors. $Qe_1$ can have $|A| = 6$ values, $Q e_2$ can have $|A|-2 = 4$ values (since $Q$ is injective).
Finally, since $Q$ is injective there are two remaining possibilities. If $Q$ is proper, then only one possibility remains for which $\det Q = 1$.
Hence there are 48 possible rotations of which only 24 are proper.
Aside: I was curious to know if $Q$ is a rotation and $C$ is such that $QC \subset C$ would imply $QC=C$ in general. The answer is negative, for example, let $Q$ be a rotation by 1 radian, and let $C = \{Q^n (1,0)^T \}_{n=0}^\infty$. Since $\pi$ is irrational, all of these points are distinct
and $QC = \{Q^n (1,0)^T \}_{n=1}^\infty \neq C$.
A: The word "rotation" here does not mean a particular action of rotating the cube around some axis - it refers to the change in the "orientation" of the cube you get by performing such a rotation. (Note that I'm using not using the word "orientation" with its layman's definition, not its usual mathematical meaning.) So if you rotate $2\pi$ radians around an axis, even though you seem to be doing something to the cube, you have in fact not changed the orientation of the cube at all - all of the vertices are right where they started! Thus we call that the identity rotation. You are looking for the number of different final orientations of the cube you can get by rotating it so that it "looks the same" after the rotation as it did before.
