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The rooms are labelled A, B, C, D, E, F, G, X, Y as shown.

We look first at routes which visit no room more than once. We need to consider only routes which go from X to A, since each of these routes has a corresponding route which goes from X to C. For example, the route X A D E Y corresponds to the route X C D G Y.

Routes which start X A then go to B or to D. There are three routes which start X A B, namely X A B E Y, X A B E D G Y and X A B E D C F G Y. There are also three routes which start X A D, namely X A D E Y, X A D G Y and X A D C F G Y.

The condition that a gap in a wall closes once a person has travelled through it means that it is not possible to visit a room more than once unless that room has at least four gaps leading into and out of it, and the only such room is D. There are two routes which start X A and visit D twice. These are X A D G F C D E Y and X A D C F G D E Y.  So there are 8 routes which start X A and there are 8 corresponding routes which start X C, so there are 16 routes in all.
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This problem is taken from the UKMT Mathematical Challenges.