Question 152
There are $n$ ($n \gt 12$) students participating in a mathematics contest. The examination paper consists of 15 fill-in-the-blank questions. For each question, the score of a correct answer is 1 point, and no point will be awarded if the answer is wrong or left blank. After analyzing all the possible cases of score distributions of these $n$ students, one finds out that if the sum of total scores of any 12 students is not less than 36 points, then there are at least 3 students among these $n$ students who answer at least 3 identical questions correctly. Determine the smallest possible value of $n$. (Posed by Liu Shixiong)