Don't Get Caught by Professor Hadamard
Puzzle for the week of 4/6
Inspired by the Fiddler on the Proof (formerly The Riddler), X’s Puzzle Corner aims to produce a weekly puzzle for readers that enjoy math, probability, and algorithms. Please submit your solution! Solutions will be accepted until 11 pm the following Sunday after the puzzle is posted (in this case 4/13/25). While it isn’t required, I encourage you to opt to have your solution shared so that we all get the chance to see how others thought about and attempted the problem! The solution and submitted responses will be posted around Wednesday at 10 am.
I make no guarantees my solutions are correct! You are all smart people so please comment if you think I made a mistake!
This week, we're on a mission to catch some cheaters.
The notorious Professor Hadamard recently administered a four-question, True/False math test to his class of four students. Given the challenging nature of his exams, he's had issues with cheating in the past. To combat this, he developed a program that analyzes students' answers by comparing them pairwise. The program counts the number of identical and differing answers between each pair of tests. If a pair has an unusually high number of matching answers, it triggers further scrutiny.
Upon running the students' tests through his program, Professor Hadamard stumbled upon a remarkable finding: Every pair of tests had exactly two answers in common and two that differed.
Intrigued, the professor pondered the likelihood of this occurrence. Assuming each of the four students answered all four True/False questions randomly, what is the probability that every pair of students' tests would exhibit exactly two matching and two differing answers?
If Professor Hadamard's class had six students and the test comprised six True/False questions, what’s the probability each pair would have 3 matching and 3 differing answers? More generally, for N students each answering N True/False questions randomly, what is the probability that every pair of students' tests would have exactly N/2 matching answers and N/2 differing answers?
Please submit your answers here. Please ask any questions in the comments.