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.)
In case you can only open it one way, you would like the observer not even to learn which way you can open it.
This solution does not need the other person to look into the cave - the other person could be blind and still receive their proof.
Other person sends you into the cave with a piece of string attached to you. They keep hold of the other end. On your return you hand them the other end of the piece of string and they pull.
If you went through the magic door they will not be able to retract the string as it will be around the central piece between A and B. If you cheated and got to the door then turned round then they would be able to retract the string.
And I cannot cheat and tie the string to the door (to make it appear that is round the central piece) as the door has no knobs on it to tie the string to.
A nice puzzle!