Is there a way to take a tour through the exhibition that passes through each door exactly once?
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Yes, this is possible because there are only two rooms with an odd number of doors. (The 'outside' is a room as well, it has seven doors.) You have already tagged the question with graph-theory, and this puzzle is equivalent to something that started the graph theory; finding an Eulerian path, named after Leonard Euler who solved the famous Seven Bridges of Königsberg problem. The rooms in your puzzle correspond to the islands; the doors to bridges. The general solution to these kind of puzzles is to
start at a room with an odd number of doors (if there is none, you can choose any starting point you like) and pick a more or less random route through the puzzle. You only need to be careful that you don't 'cut off' certain parts of the puzzle like in the example below. This is difficult to put into words/an algorithm, but very easy to see for humans.