In planet Realmamber, the inhabitants are either Knights, who always speak truthfully; Liars, who always speak falsely; Knaves, who make statements that are alternately true and false, but in which order is unknown; or those few Rebels who do not abide by the planet's traditions.

How truthful a Rebel's statements are is unknown, except that they are not the same as those who are Knights, Liars, or Knaves. Thus, a rebel will never make just one or two statements; he will always make three or more.

A, B, C and D are four inhabitants of the planet, who are busy in a conversation when a visitor from a neighboring planet stops by and asks each of them their identity. They say:


Inhabitant A:

1. My statements are not all truthful.
2. We are overworked.
3. We are all lucky to be here.
4. We Realmamberians are favored by the gods.

Inhabitant B:

1. I agree with A's third statement.
2. Every time I see a visitor, I think maybe it is one of the gods, in disguise.
3. I am doing more than my share of the work.
4. My statements are all truthful.

Inhabitant C:

1. My statements are all truthful.
2. D's second statement is false.
3. The gods do not visit us in disguise.
4. We are all overworked.

Inhabitant D:

1. C's first statement is truthful.
2. B's third statement is truthful.
3. My statements are all truthful.
4. The gods frequently visit us in disguise.

Which one is the Knight, which one is the Liar, which one is the Knave, and which one is the Rebel?

Here's the way I went about solving it. It's not a great solution since it only proves that the solution is possible, not that it's required. I didn't bother proving that the solution is required, however, once I found that my solution matched other people's. :)



1. From A's statement 1, A can neither be a Knight nor a Liar.

I then moved down to inhabitant D, since that had the richest interpersonal information.

Suppose D is a Knight. If D is a Knight, both B and C said at least one true statement, so they have to each be a Knave or a Rebel. So A would be the Liar. This contradicts 1. So D cannot be a Knight.

Suppose D is a Liar. If D is a liar, both B and C have said at least one false statement, so they have to each be a Knave or a Rebel. So A would be the Knight. This contradicts 1. So D cannot be a Liar.

Suppose D is a Knave. If D is a Knave, his third statement would be false, so his order must go FTFT. Also, if D is a Knave, A must be a Rebel, since A is a Knave or a Rebel.

So from his false statement 1, C would be a Liar. And from his true statement 2, B would be a Knight.

So A would be a Rebel, B would be a Knight, C would be a Liar, and D would be a Knave.

Given this hypothesis, no internal contradictions emerge from anyone's statements. Therefore this solution is possible. We can assume there is only one solution. Therefore this is the solution.


... by the way, A's statements must go "T, T, T" and since he is not a Knight, his last statement must be False. Therefore, Realmamberians are not all favored by the Gods.

---

EDIT: My last statement assumes that A's statement 2 is true, given that B's statement 3 is true, but I realize this is not actually required. All we know is that A's statements 1 and 3 are true. Therefore, I think we can just say that EITHER 2 is true OR 4 is true, but not both. So either they are not overworked, or Realmamberians are not all favored by the Gods.

It would have been fun if the object of the puzzle was to work out if Realmamberians are favored by the Gods or not. It would have required less ambiguity for A's statement 2, though (unless I am making a mistake somewhere).

Edited on January 13, 2010, 1:03 pm

Posted by Sam on 2010-01-12 17:57:36