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The Rebel IV (Posted on 2010-01-30) Difficulty: 3 of 5
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.

The Realmamberians are favored by the God of Light, but he has not had any success in establishing meaningful dialogue with the inhabitants. He is aware that there are those among the Realmamberians who do not respect their traditional conventions regarding veracity. Perhaps, he might have better luck making contact with a Rebel.

The God of Light, in disguise, approaches four inhabitants, exactly one of whom is known to be a Rebel. As to the group or groups of other three speakers, little is known except that no more than one is a Knight.

The God of Light asks which one of the four is the Rebel. They each respond below:

Speaker A
  1. Two of us are Rebels.
  2. B is a Knight.
  3. B was a Rebel, but he has reformed.
  4. Bs third statement is true.
Speaker B
  1. I am neither a Liar nor a Knave.
  2. As first statement is true.
  3. D is a Knight and C is the Rebel.
  4. A is not the Rebel.
Speaker C
  1. D is the Knight.
  2. I am the Rebel.
  3. Bs second statement is false.
  4. D is neither a Knight nor a Knave.
Speaker D
  1. B is either a Knave or the Rebel.
  2. I am either a Knight or a Knave.
  3. C falsely claims to be the Rebel.
  4. As third statement is false.
The God of Light throws up his hands in frustration. It is time, he decides, for him to give up and return home to try to forget about the Realmamberians.

What group is represented by each of the four speakers?

No Solution Yet Submitted by K Sengupta    
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Solution solution with explanation | Comment 2 of 3 |
A is not a Knight, for his first statement is false -- we are told there is "exactly one" rebel among the speakers. B, then, is also not a Knight, since he asserts A's first statement is true. Further, B is not a Knave, for if he were, then both his first and second statements would be false and Knaves must alternate. B is either a Liar or the Rebel.

Now, C is not a Liar, since his third statement is true (see above.) But C cannot be a Knight -- if he were then D would also be a Knight by his first statement and we know the group has at most one knight. C is therefore either a Knave or the Rebel.

Suppose C is the Rebel. Then B's 4th statement is true. But then B can't be a liar (he told the truth), and so must also be the Rebel. Since B and C can't both be the rebel, the original supposition is incorrect and C is not, in fact, the Rebel, and by process of elimination is a Knave.

C's true 1st statement (his 3rd must be true and he's a knave) means D is a knight. But since B is either a liar or a rebel, and D claims truthfully B's either a knave or the rebel, B is the Rebel. A has now made two consecutive false statements (1 and 2) and must therefore be a liar since the sole Rebel has been accounted for.

So the groups are: A = liar, B = rebel, C = knave, D = knight.
  Posted by Paul on 2010-02-02 02:57:52
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