# Proof : cannot draw this figure without lifting the pen

This question maybe ridiculous but I always found it interesting... Here it is : (I cannot put image so I put you the link of the pictures) When I was in school I used to draw houses when I was bored :

You can draw it without lifting the pen. But then I tried to draw two houses side-to-side :

Then I realized you cannot actually do it without lifting the pen. I tried any ways I could. You end up all the time with one line missing.

That's why I was wondering whether there existed a proof that I cannot actually do it.

I heard that we have to count the total number of lines and the number of intersections and that this number tells us something but I am not sure at all.

I am a first year maths student so I haven't studied graph theory yet !

Thanks !

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.