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 _______

Charlie has suggested that the missing two statements might conceivably be the two liars, claiming they are younger than themselves.  Let's assume that is the proper interpretation of the problem, and try to solve.  Excluding those two "abnormal" statements, then, all 10 "normal" statements that can be made have been made.

The full table, showing how many "normal" 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 10 normal statements, the birth order must be K L K L K, and the normal statements they can make (in birth order sequence) are 0 3 2 1 4.  Based on the number of statements made, .

The Birth Order (and number of normal statements) must be
  Knight Abby (oldest) -- 0 statements
  Liar     Diane            -- 3 statements
  Knight Emily             -- 2 statement
  Liar     Brenda          -- 1 statement
  Knight Carol (youngest) -- 4 statements
 
This, however, cannot be the case.  Brenda the liar claimed to be younger than Emily, which is a contradiction, and the oldest girl cannot be a knight.
 
Case 2) Assume the oldest girl is a liar.  The youngest must be a liar, or else there would be more that 10 normal claims.  The birth order therefore is L K K K L and the normal statements available 4 1 2 3 0 (in birth order sequence).  Based on the number of statements made,

The Birth Order (and number of normal statements) must be
  Liar     Carol (oldest)  -- 4 statements
  Knight Brenda            -- 1 statements
  Knight Emily              -- 2 statement
  Knight Diane              -- 3 statement
  Liar    Abby (youngest) -- 0 statements
 
Now there are three contradictions, so the oldest girl cannot be a liar:
   Knight Brenda claimed to be younger than Emily.
   Knight Emily claimed to be younger than Abby.
   Knight Emily claimed to be younger than Diane.

So, our assumption that the puzzle can be interpreted to allow liars to claim they are younger than themselves is incorrect.  There is only one interpretation of the puzzle which has a solution (see my first posting)

Nice puzzle, Charlie. 

Posted by Steve Herman on 2008-04-02 16:18:29