Draw it in one go

I dont want to sound pedantic, but I think this would only count as a complete soluition if you said if and only if or just only if . By what you wrote, you are saying that $\text{\# of vertices with odd \# of edges} \leq 2 \Rightarrow \text{has eulerian path}$ But that doesn't imply right away that $\text{\# of vertices with odd \# of edges} > 2 \Rightarrow \text{doesn't have eulerian path}$

The way I answered this problem without knowing about Eulerian Path is using symmetry to reduce the number of possibilities.

Here how it goes, First, I started from any point on the circumference of the circle and I tried my best to draw the picture in one go (I failed of course), then I noticed that I will fail regardless of the direction I go (providing that I started from a point on the circumference) because the circle is symmetric.

This left me with one starting point which is the center, I started at the center going to the left and again trying my best to draw the picture in one go and as expected I failed, then I noticed that when you reach the circumference, the situation is symmetric so it does not matter if you change your initial choice of direction as long as you failed once!

The net result is that you cannot draw this picture in one go.

I am not sure if this is a rigorous argument but it worked!

But it is no where said that you cannot retrace any curve

Good point, Arnav. In fact this could be used as a trick question after everybody saying no you can reveal that there was no such restriction as you assumed.

Like I mentioned no tricks more or less how it was intended or specified.this situation what we call brain teasers in hopes to think outside the box.what I'm doing playing .to en