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Knight Inference (Posted on 2012-12-16) Difficulty: 3 of 5
A remote island consists of two types of inhabitants. The liars always lie about everything and the knights who always tell the truth.

Each of Monty, Barry, Cal and Pete are inhabitants of this island. It is known that precisely one of them is a knight and the remaining three are liars.
  1. Monty says, "If Pete and I each have a dog, then exactly one of Barry and Cal has a dog."
  2. Barry says, "If Cal and I each have a cat, then exactly one of Monty and Pete has a dog."
  3. Cal says, "If Monty and I each have a dog, then exactly one of Barry and Pete has a cat."
  4. Pete says, "If Barry and I each have a cat, then exactly one of Barry and I has a dog."
Who is the knight?

See The Solution Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

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Does Barry have a dog? (spoiler) Comment 1 of 1
All of these statements are of the form "If p Then q".  
Let's call the first 1st half of each statement (p) the basis and the 2nd half (q) the inference.

"If p Then q" is false only if p is true and q is false
"If p Then q" is true if p is false or if (p is true and q is true)

There are therefore three ways to have 3 liars and one knight. Either 
(a) all of the bases and exactly one inference are true, or 
(b) all of the inferences and exactly one basis are false, or
(c) three of the bases are true.  The 4th is false but its' associated inference is true.

Let's assume that all four bases are true (case a).  Since all 4 bases are true, we know that M and P and C have dogs, and B and C and P have cats.  If B also has a dog, then all 4 inferences are false.  And if B does not have a dog, then 2 inferences are true.  Neither works out, so we can conclude that all 4 bases are not true.

Let's assume that all four inferences are false (case b).  The 3 liars have true bases, but the knight's basis is false.  Let's further assume that all B does not have a dog.  Then from inference 1 and 4, C and P also do not, and then from inference 2, M also does not.  Nobody has a dog.  This makes Basis 1 and 3 false, which does not work out.  So, B has a dog.

Given that B has a dog and all four inferences are false (case b), we can conclude from inference 1 and 4 that C and P do also, and then from inference 2, M does also.  Everybody has a dog.  This makes Basis 1 and 3 true.  Between basis 2 and 4, one must be true and the other false.  This can only be the case if B has a cat, and then from inference 3 we can conclude that P has a cat also.  So basis 4 is true, so basis 2 is false and therefore Barry is the knight.

So this leads to one possible answer
Barry is the knight
Everybody has a dog
B&P have cats, C does not and we don't know or care whether M has a cat.  

Edited on December 16, 2012, 6:34 pm
  Posted by Steve Herman on 2012-12-16 13:22:42

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