How many cycles, $C_{4}$, does the graph $Q_{n}$ contain? 
Let $n \geq 2$ be an integer. How many cycles of length $4$ (that is, subgraphs isomorphic to $C_4$) does the hypercube graph $Q_n$ contain?

Examples: For $n = 2$, the hypercube graph $Q_2$ is a single $4$-cycle, so the answer in this case is $1$. For $n = 3$, the hypercube graph $Q_3$ is a cube, and the answer is $6$.
I was told that $n$-dimensional cube graphs, $Q_{n}$ can be represented by binary but how can that help me approach this question?
 A: How many $C_4$s pass through a given vertex $v$? There are  two edges in this cycle through
$v$. There are $n$ edges through $v$ so $\binom {n}2$
pairs of edges. Once we have these two edges there is
just one way to complete the $C_4$. Don't forget there are $2^n$
vertices overall and each $C_4$ features four of them.
A: If we label the vertices with strings of $0$'s and $1$'s, edges of the cube connect vertices differing in a single position (Hamming distance $1$).
To choose the vertices of a $4$-cycle, start with any binary number you like. Choose one position to change, and then another. To close the cycle, you're forced to change the original position back, and finally the second position back. For example,
$$
11001 \overset{\color{red}2}{\to} 
1\color{red}0001 \overset{\color{blue}4}{\to} 
100\color{blue}11 \overset{\color{red}2}{\to} 
1\color{red}1011 \overset{\color{blue}4}{\to}
110\color{blue}01 
$$
This suggests:


*

*We pick one of the $2^n$ vertices

*We pick some subset $\{i,j\}$ of positions to be changed, out of the possibilities $\{1,2,\ldots,n\}$. 


But of course, it can't be that easy: surely there are multiple ways to choose the same $4$-cycle! Looking at our example above, we could have started with any of the four vertices as a starting point, but we would have had to make the same choice of positions to change. That suggests we've overcounted by a factor of $4$.
A: Here is another way to look at it: $Q_n$ is a two-dimensional projection of an $n$-dimensional hypercube, so each distinct $C_4$ in $Q_n$ is a face of the $n$-dimensional hypercube. 
When $n\ge 3$ the $n$-dimensional cube satisfies the generalized Euler polytope formula
$$
V-E+F=1+(-1)^{n-1}.
$$
For $Q(n)$ we have $V=2^n$ and $E=n2^{n-1}$ so that
$$
F=n2^{n-1}-2^n+1+(-1)^{n-1}.
$$
As to how $Q_n$ is represented with binary strings: Each vertex in $Q_n$ corresponds with each of the possible $n+1$-digit binary string and an edge exists between two vertices if their corresponding strings differ in exactly one digit.
