Zero Knowledge in the Cave There is a cave with a large entry room. From this room lead two passageways, 1 and 2. Each of 1 and 2 branches into a myriad of smaller passages, twisting and turning through the massive rock formation. The passageways go on for miles and have never been fully explored. One of the big questions has been whether passageways 1 and 2 ever connect up. Is there a way of getting from 1 to 2? Many have searched, but none have ever succeeded. Most people believe that no connection will ever be found. At least, no one used to. Now an explorer comes to you and claims to have found a passage from 1 to 2, not a very long one, either. He will prove it to you, but to you alone. Being a secretive type, he wants no one else to know. If you accompany him to the cave, he will prove the existence of the passageway to you. But there's a problem. You carry a video camera and record everything that you see. If he shows you the existence of the passage, you will be able to show the video tape to others, and they will learn of its existence as well. Not to worry, he says. Come with me. So you enter the large entry room of the cave together. Now the simplest thing to do in order to demonstrate the existence of the connection would be for him to leave through passage 1 and return through passage 2. He could easily do this. However, your film record of the event would prove to anyone else who saw it that there was a connection. Another way must be found. The explorer tells you what to do. Following his instructions, you leave the entry room for a few minutes, while the explorer enters one of the passageways. You then re-enter the room, and loudly call out one of the passageway numbers, either 1 or 2. In a few minutes, the explorer comes out of the requested passageway. You then leave the cave and repeat the process many times. Each time, the reporter enters one of the passageways unknown to you; when you return and name one of them, he is able without fail to exit from the named passage. You reason that if there were no connection between the passageways, the only way the explorer could come out the passage that you named would be if he had gone in that same one. He would have to guess which one you were going to choose, and if he were right, he could come out that one. But you have repeated the test dozens of times. The chances that someone could guess right so often is infinitisimal. The only logical explanation is that the passageway does exist. Excited, you return to the tavern where you met the explorer and show the other patrons your tape. But to your surprise, they just laugh. They don't deny that the tape is real, that the explorer did come out of the passageway you named. But they don't believe in the connection. Instead, they claim you are in league with the explorer in an attempt to perpetrate a fraud. You have simply predetermined together the sequence of numbers you would call out. Each of you has memorized the sequence, and so each time the explorer is able to anticipate the number you will call next. He enters that passage and is able, after a suitable pause, to exit from that same one when you call its number. You leave the bar, frustrated. You are convinced that the connection exists, but even though the tape shows all of the evidence that was so convincing to you, no one else finds it persuasive. The explorer has achieved his goal of proving the existence of the connection to you and you alone. Questions for the student: 1. How could you have done things differently, to produce a tape that would be convincing to others? 2. What counter-measures and conditions could the explorer have put in place to prevent you from getting a convincing tape in this manner?