Problem:
A math professor stands up in front of a room containing 100 very smart math students and says, "Each of you has to write down an integer between 0 and 100, inclusive, to guess 'two-thirds of the average of all the responses.' Each student who guesses the highest integer that is not higher than two-thirds of the average of all responses will receive a prize." If among all the students it is common knowledge that everyone will write down the best response, and there is no communication between students, what single integer should each of the 100 students write down?
Solution
Solution:
Since the average cannot be greater than , no student will write down a number greater than . But then the average cannot be greater than , and, realizing this, each student will write down a number no greater than . Continuing in this manner, we eventually see that no student will write down an integer greater than , so this is the answer.
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