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 _______
There were 10 statements made that "I am younger than ____", and only two more that could be made, for a total of 12.
A statement of this form can only be made by a younger knight, or an older liar. If the oldest girl is a knight, she can make 4 statements. The middle girl (by birth order) can always make two statements.
The full table, showing how many statements can be made is as follows:
Birth
Order Liar Knight
---------- ----- ------
1 (oldest) 4 0
2 3 1
3 2 2
4 1 3
5 (youngest) 0 4
Case 1) Assume the oldest girl is a knight. She can make 0 statements. In order to get 12 statements, the birth order must be K L L K K, and the statements they can make (in birth order sequence) are 0 3 2 3 4. Abby has made 0 statements, so she is the oldest and a knight. Carol has made 4 statements, so she is the youngest and a knight. Diane has made statements inconsistent with Carol and Emily, so Diane is a Liar and Emily is the last Knight. And then Brenda is the last Liar.
The Birth Order must be
Knight Abby (oldest)
Liar Diane
Liar Brenda (who can also claim to be younger than Carol)
Knight Emily (who can also claim to be younger than Brenda)
Knight Carol (youngest)
This works, and is a solution. Are there any others?
Case 2) Assume the oldest girl is a liar. The youngest must be a knight, or else the birth order would be L K K K L and the statements available 4 1 2 3 0 = only 10. The statements available were therefore 4 ? ? ? 4, and the only birth order that works is L K K L K, with statements available = 4 1 2 1 4 = 12. The only girls with one statement available are Abby (which accounts for one of the available statements that has not been made) and Brenda, so Emily is the middle knight (birth order 3) who can make two statements. Diane has made statements inconsistent with Carol and Emily, so Diane is a Liar and Carol is a Knight. Since Carol is a Knight, she is the youngest. The Oldest is a liar who could make 4 statements, and with only one statement unaccounted for, it must be Diane. Since Emily is younger than Abby, this only leaves the number 4 spot for Liar Brenda. But Brenda claims to be younger than Emily, WHICH IS TRUE! So, we have a contradiction, and the oldest girl cannot be a liar.
So there is a unique solution.
Nice puzzle, Charlie.
Edited on April 2, 2008, 7:46 am
I claim to be younger than Charlie (but I lie)
