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Multiple-Choice Game

There are four partners in a particular law firm. Each partner is an expert in at least one of three fields: criminal law, worker’s compensation, and patent law. These are the only areas of law that the partners of the firm practice.
 
D and F both practice in at least one of the same fields.
 
D practices in worker’s compensation and patent law.
 
F practices in only two fields.
 
D and E do not practice in the same field.
 
F and H do not practice in the same field.
In this game, the goal is to assign one or more characteristics (fields of practice) to each element (partner). Hence, it is a multiple-choice game, and therefore an elimination grid is unwarranted.

The diagram for this game will consist of four compartments, one for each of the partners D, E, F, and H:

 

Let the letters C, W, and P stand for “practices in criminal law,” “practices in worker’s comp.,” and “practices in patent law,” in that order. Placing the condition “D practices in worker’s comp. and patent law” on the diagram gives

 

Next, the condition “D and E do not practice in the same field” means that E practices only criminal law, and D practices only worker’s comp. and patent law—otherwise they would practice in some of the same fields. Adding this to the diagram yields

 
Next, the condition “D and F both practice in at least one of the same fields” means that F must practice in either worker’s comp. or patent law, or both. Adding this to the diagram along with the condition “F practices in only two fields” gives

 

Finally, the condition “H and F do not practice in the same field” means that if F practices worker’s comp., then H does not; and if F practices patent law, then H does not. In other words, ~W or ~P. In the diagram, we use an arrow to indicate this conditional relationship between F and H as follows:*

 

This diagram is a bit more restrictive than the situation warrants: F could practice criminal law. A more precise diagram would be
 

 

However, the previous diagram is sufficient for answering the questions that follow.
 
Example-1
Which one of the following must be false?
  1. F practices in exactly two fields.
  2. H practices in exactly one field.
  3. E practices in more than one field.
  4. E practices in only one field.
  5. D practices in exactly two fields.
Solution
The diagram clearly shows that (A), (B), (D), and (E) are true and that (C) is false.
 
Thus the answer is (C).
 
 
Example-2
The people in which one of the following pairs could practice in exactly the same fields?
  1. (A)   D and H
  2. (B)   E and F
  3. (C)   D and E
  4. (D)  E and H
  5. (E)   H and F
Solution
From the diagram, we see that D and H cannot practice in exactly the same fields because D practices in two fields, whereas H practices in only one.
 
This dismisses (A).
 
A similar analysis dismisses choices (B), (C), and (E). =
 
As a matter of test taking strategy this would be sufficient to mark the answer (D), but it is instructive to work out a possible assignment.
 
You should verify that the following diagram is consistent with all the initial conditions:
 

 
 
Example-3
If the combination of fields in which F practices is different from any of the combinations in which her colleagues practice, then which one of the following must be true?
  1. H does not practice patent law.
  2. F does not practice patent law.
  3. H does not practice worker’s compensation.
  4. F practices criminal law.
  5. F and H practice in the same fields.
Solution
From the diagram, we know that F must practice either worker’s comp. or patent law; but because of the new condition, she cannot practice both—otherwise she would practice in the same fields as D.
 
So F must practice criminal law, and the answer is (D).
 
 
Example-4
If a new partner who practices in exactly two fields joins the firm, then he cannot practice in all of the fields that the combination
  1. D and F do
  2. E and H do
  3. E and F do
  4. D and H do
  5. F and H do
Solution
Again from the diagram, we see that F and H practice in mutually exclusive fields. Furthermore, F practices in two fields and H practices in one field, so between them they practice in all three fields. But we are told that the new partner practices in only two fields. Hence, he cannot practice in as many fields as do F and H combined.
 
The answer is (E).
 

 

You probably have noticed that once the diagram has been constructed, assignment games are somewhat manageable. However, the diagram may not be easy to construct, and it may require considerable inspiration to figure out what kind of diagram to use. As you work the exercises in this section, you will develop more intuition in this regard.




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