The Billionaire's Gates

An unnamed billionaire software magnate has commissioned you to design his tool shed. The tool shed has five rooms. Each room connects to every room it borders and to the outdoors by at least one doorway. The layout is shown below.

The toolshed has some interesting technological features: each doorway is equipped with an infrared motion sensor gate, which turns on or off some gadget. One door might turn on and off the 36" flat-screen monitor, another the Dolby surround sound system, and yet another would activate and deactivate the external floodlights.

The toolshed is almost near completion and your client has a question for you: Assuming all the gadget's are off, is there a path one could follow through the shed which would activate every gadget, without ever deactivating one?