So this is a little puzzle that was posed to me by someone while I was sitting at a coffee shop.
The problem is to start anywhere and draw a continuous line/curve such that you cut through every line segment in the figure exactly once.
To clarify what is meant by line segment: regions A and C are bounded by 4 segments, regions B, D and E by 5 segments. Also, the line being drawn is permitted to go outside the figure into the open space marked as region F.
Drawing such a curve is impossible!
Look at regions B, D and E. They are enclosed by an odd number of line segments (5 each). Now say we were to start drawing our line outside region B, simply because B has an odd number of entrances/exits, if we were to cut through them all exactly once then our line would have to end inside region B. Similarly if we started inside B we would have to end outside B. This reasoning also gives us the same criterion for D and E. This tells us that if we were to start in B then we must end in D and E, which is clearly impossible. Similarly, if we didn't start in B we must end there but that means we have to start in D and E, again clearly impossible.
Lets try togeneralize the problem and introduce some abstraction. First, clearly the only thing that matters in the problem is the number of borders/line segments a region is enclosed by and which other regions it connects to through these borders. So lets represent each region by a dot or a point which we will call a vertex and we draw a line/arc between two vertices for every border connecting the two regions with each other, we'll call these lines/arcs edges. Call these new figures/configurations we get graphs. So now our general problem becomes that for a given graph when can we start at some vertex and come up with a path that traverses every edge exactly once? Well, as we noted earlier the thing that really made our impossibility proof work was oddness/eveness of the number of borders that a region has. So correspondingly lets call the number of edges coming out of a vertex the degree of the vertex.
The general problem in it's abstract form has a complete solution/proof (the proof essentially exploits the whole even-odd thing to it's fullest), and if you would like to learn more, a good place to start would be here.