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(The complete setup for this game can be found here: lsat/viewtopic.php?t=13934)
The correct answer choice is (E)
This type of question would normally be time consuming in the typical game. However, because we have all of the numerical information about the illnesses, and a significant amount of information about the symptoms of each illness, it is easier than would normally be expected. Some of the answers—such as (B)—can be eliminated with a quick glance at the original diagram.
However, there is a better way to think about this problem, and that is from an abstract standpoint. One way to force two illnesses to have exactly one symptom in common is to choose an illness with all three symptoms, and then pair it with and illness with exactly one symptom. This ensures that the two illnesses have exactly one symptom in common. Because M has all three symptoms, M must have at least one symptom in common with all other illnesses. Thus, it automatically has exactly one symptom in common with both K and N (although not necessarily the same symptom). M and K do not appear as one of the answer choices, but M and N do appear in answer choice (E). Because N only has one symptom it must be one of the symptoms that M has. Thus M and N have exactly one symptom in common and answer choice (E) is correct.
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