Answer to Problem of the Week for 06/11/2001

Answer to June 11, 2001 Problem

Contributed by Warren Buckner

The Island Problem

Professor Wise finds himself investigating a strange island known only as The Island of Implausibility. On this island there are two tribes of people none of whom shows any indication of which tribe they belong to except for the following implausible fact: members of the Friendless Tribe always state what is true and members of the Untrusted Tribe always state what is false. The only things they agree upon are two rules that had come down to them from the distant past: a stranger to the island can only ask one question of any native of the island, and, the question can only be answered "yes" or "no"; under penalty of death. Professor Wise unwisely tried to inquire of a native more than one question and was now being chased to pay the penalty. He came upon a fork in the road, guarded by one native of the island. A sign was there that said "This way to the shore" and "This way to quicksand". Each sentence had an arrow with it. Clearly the sign was meant to be at the fork indicating which road led where, but the sign had been knocked down and lay on the ground, thus giving no information. The shore meant his boat and freedom. Turning around meant meeting his pursuers. The other road meant death in quicksand. He had but one quesiton he could ask the guard. What question should he ask?
Solution to Problem: There is more than one question that will do, but they basically are variants of: Pointing to a road, Prof. Wise asks: "If I asked you "Is this the road to the shore?" would you say "yes"? In other words, you must embed a question within a question (so that both the liar ands the truth sayer will answer the same way). ==================================== Assume Prof. Wise points to road A and Road B is to the shore. Friendless, telling the truth, will say "no". Untrusted, who always lies, will also say "no" (he would say "yes" if asked if it is the road to the shore (a lie), so lying here he says he would not say "yes". If Prof. Wise points to road A and Road A is to the shore, both Friendless and Untrusted would say "yes". Thus "no" means take Road B and "yes" means take road A. Symmetrically the same argument would hold if you reverse the road pointed to. "No means take the other road, "yes" means take the road pointed to. As a side note, might this be an interesting way to point out how the negative of a negative is a positive?

Professor Wise finds himself investigating a strange island known only as The Island of Implausibility.

On this island there are two tribes of people none of whom shows any indication of which tribe they belong to except for the following implausible fact: members of the Friendless Tribe always state what is true and members of the Untrusted Tribe always state what is false. The only things they agree upon are two rules that had come down to them from the distant past: a stranger to the island can only ask one question of any native of the island, and, the question can only be answered "yes" or "no"; under penalty of death.

Professor Wise unwisely tried to inquire of a native more than one question and was now being chased to pay the penalty. He came upon a fork in the road, guarded by one native of the island. A sign was there that said "This way to the shore" and "This way to quicksand". Each sentence had an arrow with it. Clearly the sign was meant to be at the fork indicating which road led where, but the sign had been knocked down and lay on the ground, thus giving no information.

The shore meant his boat and freedom. Turning around meant meeting his pursuers. The other road meant death in quicksand. He had but one quesiton he could ask the guard. What question should he ask?

Solution to Problem:

There is more than one question that will do, but they basically are variants of:

Pointing to a road, Prof. Wise asks: "If I asked you "Is this the road to the shore?" would you say "yes"?
In other words, you must embed a question within a question (so that both the liar ands the truth sayer will answer the same way).
====================================

Assume Prof. Wise points to road A and Road B is to the shore. Friendless, telling the truth, will say "no". Untrusted, who always lies, will also say "no" (he would say "yes" if asked if it is the road to the shore (a lie), so lying here he says he would not say "yes".

If Prof. Wise points to road A and Road A is to the shore, both Friendless and Untrusted would say "yes".

Thus "no" means take Road B and "yes" means take road A.
Symmetrically the same argument would hold if you reverse the road pointed to.
"No means take the other road, "yes" means take the road pointed to.
As a side note, might this be an interesting way to point out how the negative of a negative is a positive?

Correctly solved by:

1. Keith Mealy	Cincinnati, Ohio
2. Renata Sommerville	Austin, Texas
3. Walt Arrison	Philadelphia, Pennsylvania
4. Ricki Stern	Highland Park, New Jersey
5. Richard K. Johnson	La Jolla, California