Maybe this will help - whenever you're testing values on DS, have this goal in mind: "let me pick some values that will give me a YES answer, then some that will give me a NO answer." This way, you're thinking about extremes, and not just testing randomly.
(1) b > c > 69
So let's think about extremes - what are the smallest possible values of b and c? 71 and 70. If that's the case, our set is (rearranged in order except for a): [a, 51, 70, 71, 72, 85]. Here, it doesn't matter what a is; the median has to be between 70 and 72. This would give us a NO answer.
So can we come up with a YES answer? What would the opposite extreme be? We just know that b and c are greater than 69, so there's no limit to how big they could be! We could say c = 1000, b = 1001, and a = 1002. In that case, our set would be: [51, 72, 85, 1000, 1001, 1002]. Clearly the median is way bigger than 80, so we get a YES answer.
If we have a YES and a NO, it's insufficient.
(2) a < c < 71
Let's think of extremes again. What are the smallest possible values? Let's say a = 1, b = 2, and c = 3. Our set is [1, 2, 3, 51, 72, 85]. Clearly the median is less than 80, so we get a NO.
What are the largest possible values? The largest that c can be is 70, and the largest 1 can be is 69. There are no restrictions on b, so we can say b = 1000. Our set is [51, 69, 70, 72, 85, 1000]. Here, even with a and c at their highest, the median is still only 71. We can't find any values where the median is greater than 80. This is SUFFICIENT.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
