BRAIN PUZZLE
>> 13 January 2009
Do you have powers of logic? Test yourself with this little puzzle:
You are visiting a certain island, all of whose inhabitants can be divided into two groups: the knights and the knaves. Anything a knight says is true and anything a knave says is false. You meet two inhabitants of this island, Boris and Amanda. You don't know to which group either Boris or Amanda belongs, but you ask Boris what Amanda would say if she were to say whether Boris was a knight or a knave. Boris replies, "Amanda would say that I am a knight." Based on just this information, can you tell whether Amanda is a knight or a knave? What about Boris?
Cast your votes below and explain your reasoning in the comments section.
7 comments:
Assume Amanda is a knave. She's either going to say "Boris is a knight" or "Boris is a knave".
If she says "Boris is a knight", then Boris is a knave. And if Boris is a knave then his statement "Amanda would say that I am a knight" must be false - ie it is impossible that she says "Boris is a knight". Does not compute.
If she says "Boris is a knave" then Boris is a knight. And if Boris is a knight then his statement "Amanda would say that I am a knight" must be true - ie it is impossible that she says "Boris is a knave". Does not compute.
Therefore our assumption that Amanda is a knave must be false - she is a knight!
Based on the statement "all of whose inhabitants can be divided into two groups" I'm going to go out on a limb and argue that because "Boris and Amanda" are "inhabitants", they can be divided into two groups. Therefore one must be a knight, and the other a knave. Amanda is a knight so Boris must be a knave. Eh? Eh? Any takers?
For another brain puzzle (not nearly as challenging) try to figure out Boris' password. LOL
All of this depends on whether they are from different camps. If they have to be from different camps, then my rationale follows Phillip's.
Amanda is the knight, Boris is the knave. Boris is lying in saying that Amanda would say he is a knight.
There are 4 possible scenarios (the premise that Boris and Amanda must be different was not stated in the puzzle, and so cannot logically be used):
1. Boris and Amanda are both knights.
2. Boris and Amanda are both knaves.
3. Boris is a knight and Amanda is a knave.
4. Boris is a knave and Amanda is a knight.
Scenario 1 is valid, as if Amanda is telling the truth about Boris, and Boris the truth about Amanda, each will state that the other is a knight. Scenario 2 is invalid, as if Amanda states that Boris is a knight, she would be lying, but Boris would be telling the truth about Amanda's response. Scenario 3 is also invalid, as Amanda would be have to be telling the truth (as a knave). Scenario 4 is valid, as if Amanda is telling the truth about Boris, she will call him a knave, but Boris must lie about this and say that she would call him a knight.
As scenarios 1 and 4 are each valid, Amanda must be a knight, but the identity of Boris cannot be determined.
First, assume that population of the island > 2.
Assume we use the following substitution:
Boris = T
Amanda = F
Knight = T
Knave = F
Say = ->
is = "="
then,
Case 1; B=T and A=T
Then, A->B=T and B->(A->B=T)
Conclusion: Valid assumption
Case 2; B=F and A=T
Then, A->B=F and B->(A->B=T)
Conclusion: Valid assumption
Case 3; B=T and A=F
Then, A->B=F and B->(A->B=F)
Conclusion: Invalid assumption
Case 4; B=F and A=F
Then, A->B=T and B->(A->B=F)
Conclusion: Invalid assumption
Therefore: Only case 1 and 2 were possible.
Then,
Case 1: B=T and Case 2: B=F -> Contradiction
Case 1: A=T and Case 2: A=T -> Agreement
Therefore:
Boris = Inconclusive
Amanda = T = Knight
Then I became an idiot and miss-clicked and said both were inconclusive.
p.s. I deleted my last comment because the format looked terrible. :p
They are either both lying, making them both knaves, or they are both telling the truth, making them both knights. Either way, you don't really have a way to tell.
Huh... I supposed I should have said Boris=B and Amanda=A instead!
Post a Comment