Arlene(A), Brenda(B), Cheryl(C), Daniel(D), Emmett(E) and Farley(F) stayed in a hotel.

1) Each stayed in a different one of six rooms as shown here, identified by his initials :

         +----+----+----+----+
         |    | C  |    | E  |
         | B  +----+ D  +----+
         |    |    |    |    |
         +----+    +----+    +
         |      A  |      F  |
         +---------+---------+

2) One of the six murdered one of the other five.

3) If the murderer and the victim stayed in rooms that did not border on each other, then Arlene or Farley was the victim.

4) If the murderer and the victim stayed in rooms that bordered on different numbers of rooms, then Brenda or Cheryl was the murderer.

5) If the murderer and the victim stayed in rooms that were different in size, then Daniel or Emmett was the murderer.

Who was the murderer? Who was the victim?

I am always amazed and impressed that people can arrive at an answer to this type of problem by simply thinking it through. Solutions like: If A = B and C = D then it cant be E, but if thats the case A must = E so it cant be A..., are almost as difficult for me to follow as trying to solve the problem this way. I tend to use the Brute Force Method which is used below to solve this problem.

There are three statements made about the murderer/victim (Statements 3, 4 and 5 of the problem).

STEP 1 - First I list all of the possible True/False combinations of these three statements:

3 4 5
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F

STEP 2 - Then I list all of the True/False murderer/victim combinations for each statement:

Statement 3 - The murderer/victim stayed in rooms that did not border each other.
TRUE  FALSE
A,E    A,B
B,D*  A,C
B,E*  A,D
B,F    A,F
C,E*  B,C
C,F    C,D
         D,E
         D,F
         E,F
     
Statement 4 - The murderer/victim stayed in rooms that bordered on different numbers of rooms.
TRUE  FALSE
A,B   A,D
A,C   B,E
A,E*  C,F
A,F*
B,C
B,D
B,F
C,D
C,E
D,E*
D,F*
E,F*

Statement 5 - The murderer/victim stayed in rooms that were different sizes.
TRUE  FALSE
A,B*  A,F
A,C*  B,D
A,D    C,E
A,E
B,C*
B,E
B,F*
C,D
C,F*
D,E
D,F
E,F

STEP 3 - start eliminating
Because of the addition info provided, I can immediately eliminate some combinations from the True side of each statement listed in STEP 2 (marked with a *):
Statement 3 - Eliminate all combinations from the True side that do not contain Arlene or Farley.
Statement 4 - Eliminate all combinations from the True side that do not contain Brenda or Cheryl.
Statement 5 - Eliminate all combinations from the True side that do not contain Daniel or Emmett.

Now I need to eliminate the T/F combinations of statements listed in STEP 1 that are invalid:
T,T,T and F,T,T can immediately be eliminated because if both statemenst 4 and 5 are true, there is a conflict as to who the murderer is.

For a statement T/F combination to be valid, there must be at least one murderer/victim combination that is common to all three statements.

Lets look at TTF. A,E has already been eliminated from the True side of Statement 3 so A,E is not common. B,F does not exist on the False side of Statement 5 so B,F is not common. That leaves C,F which does not exist on the True side of Statement 4 so C,F is not common. So, TTF is invalid.

Keep working through the remaining five scenarios and you will find that the only scenario that has a common murderer/victim combination is the FFT scenario and that common combination is A,D (Arlene and Daniel).

Because Statement 5 is True, Daniel must be the murderer and Arlene the victim.

Posted by Bender on 2005-08-31 04:08:53