The Billionaire's Gates Revisited

[Ghost Post]

Every room had a doorway to all the adjacent rooms, plus one to the outside. This billionaire wanted to be able to walk through every doorway exactly once- not missing any doorway, or going through any doorway twice.
- Quoted exactly
From the outside enter room D
From room D enter door on adjoining wall to room E
Exit room E through outside door
Enter room C through outside door
From room C enter door on adjoining wall to room E
From room E enter door on adjoining wall to room B
From room B enter door on adjoining wall to room D
From room D enter door on adjoining wall to room A
Exit room A through outside door
Enter room B through outside door
~Each doorway passed through only once each
~Just one of the ways it could be done

[Ghost Post]

Well it seems that I have forgotten two doors.

[CrystyB]

That was the whole problem with the INITIAL gates: within any Euclidian geometry, if you have a graph with more that two nodes (vertex) with an odd number of arcs (doors in this case), you CAN'T pass through it like the problem wants.