TITLE: There are apparently $3072$ ways to draw this flower. But why?
QUESTION [177 upvotes]: This picture was in my friend's math book: 
 
Below the picture it says: 

There are $3072$ ways to draw this flower, starting from the center of
  the petals, without lifting the pen.

I know it's based on combinatorics, but I don't know how to show that there are actually $3072$ ways to do this. I'd be glad if someone showed how to show that there are exactly $3072$ ways to draw this flower, starting from the center of the petals, without lifting the pen (assuming that $3072$ is the correct amount).

REPLY [12 votes]: Others have already solved the problem. I just wanted to add that, properly speaking, this problem and/or its solution belong to a branch of math called graph theory. Formally:

Proposition. The following undirected graph has $3072$ directed Eulerian trails starting at $x$ and ending at $z$.


We could be even more precise by describing the graph in more formal terms, of course, rather than just drawing a picture. This can be done by giving a symmetric function $\{x,y,z\}^2 \rightarrow \mathbb{N}$ that tells us how many edges go between any two vertexes. In this particular case, our function is