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.

(In reply to What am I missing? by pcbouhid)

If the stranger must stay in the entrance, how can you prove that the door was not already open when you return, supposedly passing from side A to side B? 

How can you prove there is not another passage between side A and B?

How can you prove there is a door?

Does the stranger see the bifurcation?

If the stranger knows the cave as you do, knows the magic door is closed before you offer your proof, can see the bifurcation, and knows there is no one else in the cave, it should only require that he sees you walk down one side and return from the other. 
 

Posted by Dej Mar on 2008-09-15 00:42:58