How many rotations are there in $\mathbb R^3$ which take $C$ to itself? Let $C$ denotes the cube $[-1,1]^3\subset\mathbb R^3.$ How many rotations are there in $\mathbb R^3$ which take $C$ to itself?
A. $6$
B. $12$
C. $18$
D. $24$

Let $H$ be the set of vertices of the cube, i.e. $H=\{1,2,\cdots,8\}$. Label the vertices of the cube $H$ such that $1$ is adjacent to $2$ and so forth. Consider the rigid motions (that is rotations) of vertex $1$. 
There are how many different vertices to which $1$ can be sent, not including the motion which fixes the cube?

 There are $7$.

So including the one that fixes the cube, there are .... total. Then there are how many possibilities for the placement of vertex $2$ as it must be adjacent to vertex $1$. 

 There are $3$.

This yields a total of how many total possible rigid motions of the cube?

 $3\cdot 8=24$ total ways.

Since a rigid motion must fix the edge connecting vertex $1$ and vertex $2$, this is sufficient to determine the cube. Hence, there are .... total rigid motions (rotations) which fix the cube.
