There are 100 lockers in a hallway, all closed. 100 students walk by one at a time. The first student opens every locker. The second student closes every 2nd locker. The third student toggles every 3rd locker. This continues for all 100 students. Which lockers are open at the end?

Why this works

At first glance, this riddle might seem like a perplexing puzzle of chaos, but it reveals a fascinating pattern rooted in mathematics. The key lies in understanding how the lockers are toggled open and shut with each student's actions. Each locker is opened or closed based on its divisors; for example, locker number 12 is toggled by students 1, 2, 3, 4, 6, and 12. This means a locker will end up open only if it has an odd number of total divisors, and the only numbers that meet this criterion are perfect squares. Why? Because divisors typically come in pairs (like 1 and 12, 2 and 6), but a perfect square, such as 9, has a central divisor that repeats (3 in this case), resulting in an unpaired divisor. This riddle not only entertains but also showcases the beauty of numbers and their properties. It invites us to explore the world of mathematics in a playful way, leading to that delightful “aha moment” when the realization dawns that only the lockers numbered with perfect squares—1, 4, 9, 16, and so on—remain open. This clever twist highlights how seemingly simple actions can lead to intricate outcomes, revealing deeper truths about the nature of numbers. Fun fact: this kind of riddle is a classic example of a mathematical problem that has been enjoyed for centuries, often found in recreational mathematics, where logic and number theory combine to create engaging challenges!