You have just finished exploring a huge cave with a long tunnel that eventually connects with itself, with a magic door in the middle. The magic door has no knob -- instead it requires a secret password to open. When it is closed, the cave can be thought of as an entrance tunnel with two tunnels (tunnel A and tunnel B) branching off of it. These are shown below:

(A map of the cave)

    | |
    | |
    | |
 ___| |___
|  _____  |
| |     | |
| |     | |
| |_____| |
|    #    |
|____#____|

You tell your claim to another person, who is interested but wants proof that you know the secret. You want to show you know, but don't want to share the secret with a stranger. How can you prove to him beyond a reasonable doubt that you know the secret password?

(Assume the other person must stay in the entrance tunnel of the cave.)

Note: In case you can only open it one way, you would like the observer not even to learn which way you can open it.