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The Billionaire's Gates Revisited

It is possible to map the toolshed onto the surface of a torus (or "doughnut," to use the technical term) as shown in the illustration below. You can see that it should be possible to cross each boundary exactly once on a single trip.

But by moving Room B around the center of the torus, we've actually ended up removing one of the rooms. Room B and the outside are now the same region-- just walk through the doughnut hole.

The billionaire's original goal was to have a sensor at each doorway that would activate/deactivate a gizmo. So while the toolshed's geometry on the asteroid is the same, number of distinct boundaries has changed.

Is it possible that a different geometry could be used?

No, for the same reason as in the original puzzle: in order to pass through all the doors in a given room on a trip, if a room has an odd number of doors, you must either begin or end in that room. Rooms A, B and C all have an odd number of doors, so no solution exists.

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