b)Every evening, Rector Tissaia goes to all the rooms to extinguish the candles, but of course she wants to be as efficient as possible and not enter any room twice.1 Can the Rector find such a way? Justify your answer
Note: Translate the question into graph theory terms first.
On
Yes. Just draw the graph that arises from the picture (I am assuming that the shapes that are outlines coincide with the shapes that are not outlines). Each "room" is a node, and each coinciding shape is an edge that connects each coinciding node (or room). Then, see if there is an Euler Circuit. The criteria for an Euler Circuit is that all nodes are even degree, as in each node is connected to an even amount of edges. If there is any node connected to an odd amount of edges, then there is no Euler Circuit, so Rector cannot do this.
Once you draw the graph in accordance to my instructions from the comments, you will quickly see that this is impossible. If there is a route without repeating rooms, then clearly 8 should be the start and 9 the end. Then you see that you have to get to room 2 from room 3, but from room 2 you can only get to 10 and since 10 is the only way to get to 9, the end of the route must be 3-2-10-9. But given that from 11 you can only get to 10 you have a problem.