Problem of rooms

Question

A rectangle is divided into some smaller rectangles.Each two adjacent rectangles share a door which connects them.Prove that we can start from one of the small rectangles and pass them all without crossing a rectangle more than once.

Answer 1

Here's a tip in the right direction. The main difficulty of an inductive argument would be the following hypothetical case:

Here, it is not obvious how to adapt the pre-existing path to to include this new rectangle. So, why not make the inductive hypothesis be that there exists a path which never exits a rectangle, $r$, from the same "side" that $r$ was entered from, avoiding this problem altogether. With that inductive hypothesis, here’s the case work:

Hopefully everything is clear now?

Answer 2

Convert the rectangular structure to a graph, where each door becomes a vertex and each room becomes an edge. Since each door connects exactly two rooms, each vertex is therefore of degree two. Since no vertex is of odd degree, a Eulerian cycle is possible, i.e. each room/edge/bridge can be visited without repetition.