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Mentor Exercise

 
Directions: The following group of questions is based on a set of conditions. Choose the response that most accurately and completely answers each question. Hints, insights, and the answers are provided in the right-hand column.

Eight people—Adam, Bob, Carrie, Dan, Elaine, Fred, Greg, Hans—are seated around a square table. There are two people to a side, and each person sits directly opposite another.
 
Bob sits directly opposite Hans.
 
Adam sits immediately between Greg and Fred.
 
Although this game involves a square, all the properties derived for circles still hold. The condition “Bob sits directly opposite Hans” is naturally symbolized as B<—>H, and the condition “Adam sits immediately between Greg and Fred” can be symbolized as GAF (The flip-flop symbol will not be explicitly written, just remember that G and F can be interchanged.) The diagram will consist of a square:
 

The condition B<—>H forms the base axis. Since the properties of circular diagrams hold for square diagrams, we can place the base axis anywhere on the diagram. Place it at the top as follows:
 

The base group GAF cannot be placed on the diagram at this time. Since there is no mention of orientation of the square, we don’t have to consider the diagram’s mirror image, i.e., the diagram obtained by interchanging B and H (See, however, Questions 4 and 5).
 
Question-1
If Elaine does not sit next to Bob, which one of the following must be true?
  1. Adam or Greg must sit next to Hans.
  2. Greg must sit directly opposite Elaine.
  3. Either Carrie or Dan must sit next to Bob.
  4. Adam must sit next to Hans.
  5. Fred must sit next to Bob.
Solution
There are two possible positions for the condition GAF:
 
 

Note the flip-flop of G and F is not needed in this problem. Use these two diagrams to deduce that the answer is (C).
 
 
Question-2
If Dan sits directly across from Adam, and Elaine cannot sit next to Bob, then which of the following people could sit next to Bob?
  1. Dan
  2. Carrie
  3. Fred
 
  1. I only
  2. II only
  3. III only
  4. I and II only
  5. I and III only
Solution
There are two possible places for D:
 

We reject the second diagram since it forces E next to B, violating the supplementary condition ~(EB).
 
Be careful: The flip-flop of G and F is needed for this question.
 
The answer is (E).
 
 
Question-3
If Carrie does not sit next to Bob, which one of the following must be false?
  1. Hans sits next to Fred.
  2. Adam sits opposite Carrie.
  3. Greg sits next to Bob.
  4. Dan sits to the immediate right of Hans.
  5. Hans sits between and next to Dan and Fred.
Solution
If Hans sits between and next to Dan and Fred, then we get the following diagram:
 

But this diagram forces C next to B, violating the supplementary condition ~(CB).
 
The answer is (E).
 
 
Question-4
If Greg sits to the immediate left of Adam, then which one of the following is possible?
  1. Adam sits next to Bob.
  2. Hans sits next to Bob.
  3. Adam sits next to Hans.
  4. Fred sits next to Bob.
  5. Bob sits directly opposite Carrie.
Solution
Caution: Since this question applies an orientation to the square, you must consider the mirror image diagram:
 
 
The answer is (D).
 
 
Question-5
If Dan sits next to Elaine who sits next to Hans, what is the maximum number of distinct arrangements of the eight people around the square?
  1. 2
  2. 4
  3. 6
  4. 8
  5. 10
Solution
Caution: Although there is no mention of orientation in this question, you still must consider the mirror image diagram, since it is still a distinct order.
 
The answer is (D).
 
 
Question-6
If Greg sits directly opposite Dan, which one of the following is complete and accurate list of the people any one of whom could sit next to Bob?
  1. Greg
  2. Greg, Fred, Dan
  3. Greg, Fred, Elaine, Carrie
  4. Greg, Fred, Dan, Carrie, Elaine
  5. Greg, Fred, Dan, Carrie, Adam
Solution
There are four possible positions for D:


Use these diagrams to deduce that the answer is (D).
 




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