A man named John was trying to find gold. He went to an island where every inhabitant is either a knight or a liar. He met three inhabitants, A, B, and C. A and B made these statements.

A:We are all liars.
B:There is gold on this island.

Then, he asked one of them, "Is it true that you are a knight and there is no gold on this island?" The inhabitant answered, and he could not figure out if there was gold on the island. Then, one of the inhabitants either said that A was a liar or said that B was a liar, and John knew if there was gold on the island.

There is not enough information yet for you to figure out if there is gold on the island. However, if I told you whether the last inhabitant said that A was a liar or said that B was a liar, without telling you which inhabitant said it, then you would be able to figure out if there is gold on the island. Who did the last inhabitant say was a liar, and is there gold on the island?

If A was a knight, then they would all be liars, which is impossible. Therefore, A is a liar. Since A is a liar, they are not all liars, so either B or C is a knight. Then, John asked, "Is it true that you are a knight and there is no gold on this island?" Here are all of the possible cases.

Case 1:A answered. Since A is a liar, it is not true that A is a knight and there is no gold on the island, so A would lie and say, "Yes." It is impossible to figure out if there is gold on the island.

Case 2:B said, "Yes." Then, B is a liar because a knight would not say that there is gold on the island and also say that they are a knight and there is no gold on the island. Since B is a liar and said that there was gold on the island, John would know that there is no gold on the island.

Case 3:B said, "No." Then, B is a knight because a liar would not tell the truth and say that it is not true that they are a knight and there is no gold on the island. Since B is a knight, it is not true that B is a knight and there is no gold on the island, so there is gold on the island.

Case 4:C said, "Yes." If C is a knight, then there is no gold on the island, so B is a liar. If C is a liar, then B is a knight because either B or C is a knight. Therefore, B and C are of different types. In this case, John would not know if there is gold on the island.

Case 5:C said, "No." Then, C is a knight and there is gold on the island for the same reason as Case 3.

John could not figure out if there was gold on the island, so either Case 1 or Case 4 holds. Now, one of them either said that A was a liar or that B was a liar. No knight or liar can claim to be a liar, so either B or C said that A was a liar, or either A or C said that B was a liar. Here are all the cases for Case 1 and Case 4.

Case 1a:B said that A was a liar. Since we know that A is a liar, B is a knight, so there is gold in the island.

Case 1b:C said that A was a liar. All John would know is that C is a knight.

Case 1c:A said that B was a liar. Since A is a liar, B is a knight and there is gold in the island.

Case 1d:C said that B was a liar. All John would know is that B and C are of different types.

Case 4a:B said that A was a liar. For the same reason as 1a, there is gold in the island.

Case 4b:C said that A was a liar. Since A is a liar, C is a knight. Also, B and C are of different types, so B is a liar and there is no gold in the island.

Case 4c:A said that B was a liar. For the same reason as 1c, there is gold in the island.

Case 4d:C said that B was a liar. Then, B and C are of different types, but John already knew this.

John knew if there was gold in the island, so either 1a, 1c, 4a, 4b, or 4c holds. In 4b, there is no gold, but in 1a, 1c, 4a, and 4c, there is gold. If you knew that the last inhabitant said that A was a liar, then there could be gold or no gold. Therefore, the last inhabitant said that B was a liar, so there is gold on the island.

Comments: (
You must be logged in to post comments.)