Proof : cannot draw this figure without lifting the pen

Solution 1:

This sounds like a great excuse to study (just a little) graph theory to me! Check out the conclusions of Euler in this famous problem. It should tell you all you need to know (and more). In particular, the second figure has $4$ nodes of odd degree (that is, an odd number of segments having that common endpoint)--the outer bottom corners and the two nodes where the "houses" meet--so it cannot be traced without lifting the pen or retracing at least one segment of the path.

Solution 2:

Take a vertex with three edges. Assume you do NOT start there. Then there is a first time you will reach this vertex, through one of the possible paths leading to it. It won’t be the end of your drawing because two other paths are not yet traversed. Thus, you will leave this vertex, through a second adjacent path. As a result, when you traverse the third path some time, you won’t be able to leave this vertex because no untraveled paths will be available. In other words, your drawing will have to end here.

So, we showed that if you don’t start at a vertex (with odd number of edges), you will have to end there.

This is similarly true about vertices with 5 edges. Thus, if any graph has more than two vertices with odd degrees, you won't be able to draw it.