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 Who's Younger? (Posted on 2008-04-01)
Abby, Brenda, Carol, Diane and Emily are all of different ages. Two are liars and three are knights (knights always tell the truth; liars always lie).

Brenda claimed to be younger than Emily.
Carol claimed to be younger than Abby.
Carol claimed to be younger than Brenda.
Carol claimed to be younger than Diane.
Carol claimed to be younger than Emily.
Diane claimed to be younger than Brenda.
Diane claimed to be younger than Carol.
Diane claimed to be younger than Emily.
Emily claimed to be younger than Abby.
Emily claimed to be younger than Diane.

There were two other, similar, statements made, besides the ones metioned above, and in fact those were the only other statements like that that could be made.

That might be a little ambiguous, but resolving this ambiguity is part of the fun, and you can do it.

Who were the two liars and what was the order of their ages?

You can even fill in the two unheard remarks:

_______ claimed to be younger than _______
_______ claimed to be younger than _______

 See The Solution Submitted by Charlie Rating: 3.0000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
 Solution Comment 7 of 7 |

The fourth and seventh statements contradict, so exactly one of Carol and Diane is a knight and one is a liar.  Similarily the eighth and tenth statements contradict, so exactly one of Emily and Diane is a knight and one is a liar.

If Diane is a knight, then Carol and Emily must be the two liars.  Then Carol is the oldest from statements 2-5.  Then Emily is the second oldest statements 1,9,10.  And Brenda is not the youngest from statement 6.

- If Abby was the youngest, then she could claim to be younger than all four of the others, but there are only supposed to be two such additional statements possible.

- If Diane is the youngest, then there are at least three more possible statements: Diane claiming to be younger than Abby, and Abby claiming to be younger than Carol and Emily.
The assumption that Diane is a knight is false.

That leaves Diane as a liar, and Carol and Emily as knights.  Carol is then the youngest from statements 2-5.

- If Brenda is a knight, then Brenda is second youngest from statements 1,9,10, Emily is in the middle, and Abby and Diane are the oldest two in some order.  With Brenda as second youngest, she could make two more claims, being younger than Abby and Diane.  For Diane to be unable to make another claim, she must be second oldest with Abby being oldest.  Since Abby must be the last liar and the oldest, she can claim to be younger than all the others.  This contradicts the limit of two addition possible statements, so the assumtion that Brenda is a knight is false.

- That leaves Brenda as the other liar, then Emily is second youngest from statements 1,9,10. And Diane is older than Brenda.  Emily can make one more claim, being younger than Brenda. Abby cannot be the in the middle, otherwise she could make two more claims.  Since Diane is older than Brenda, Brenda is in the middle.  Brenda then could lie and make one more statement, claiming to be younger than Carol.  Then for Abby to be unable to make a third possible statement, she must be the oldest.

Then, the order youngest to oldest is then Carol(K), Emily(K), Brenda(L), Diane(L), Abby(K).  The two additional statements are 'Emily claimed to be younger than Brenda.' and 'Brenda claimed to be younger than Carol.

 Posted by Brian Smith on 2008-04-02 22:27:45

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