Locker Puzzle: Which 10 Lockers Stay Open?

The Principal's Evening Ritual Meets a 100-Student Prank

Picture a school corridor lined with exactly 100 lockers, each one identical and neatly shut. Every evening, after the last bell rings and the hallways fall quiet, the principal walks the length of that corridor to make sure every single locker door is closed. It's a small ritual, a final check before heading home. One day, though, 100 students decide the principal deserves a surprise.

Their plan unfolds in perfect sequence. The first student in line walks down the corridor and opens every locker, all 100 of them. The second student follows, starting at locker number 2, and closes every even-numbered locker. The third student starts at locker 3 and toggles every multiple of 3: if it's open, they close it; if it's closed, they open it. The fourth student does the same for multiples of 4, the fifth for multiples of 5, and so on, all the way down to the 100th student. By the time the last student finishes, the corridor is a jumble of open and closed doors. The question hangs in the air: which lockers will the principal find open the next morning?

Before diving into the answer, it's worth grabbing a pencil and paper. The pattern is right there, but you have to poke around a bit to see it. Consider locker number 24. Student 1 opens it. Student 2 closes it. Student 3 opens it. Student 4 closes it. Student 5 skips it entirely. The students who touch locker 24 are exactly the numbers that divide 24 evenly: 1, 2, 3, 4, 6, 8, 12, and 24. Every time a student toggles it, the door flips. So the final state of any locker comes down to one thing: how many factors its number has.

Most numbers have factors that pair up neatly, which means an even count. But some numbers have a factor that pairs with itself, giving them an odd number of factors. Those numbers are the perfect squares. So the lockers left open are precisely the squares from 1 to 100: 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. When the principal walks in the next morning, those ten doors will be standing wide open, a quiet math joke that 100 students pulled off together.

Tracking the Toggles: The Locker Puzzle's Hidden Pattern

The key to cracking the locker puzzle is to stop thinking about what each student does and start thinking about what happens to a single locker. Take locker 24. It gets touched by student 1, who opens everything. Student 2 comes along and closes it. Student 3 opens it again. Student 4 closes it. Students 5 and 7 walk right past it. But students 6, 8, 12, and 24 all take their turn flipping it. That is eight total touches.

Those eight students share one thing in common: their numbers are the divisors of 24. Every locker gets toggled exactly once for each of its factors. And just like that, the puzzle stops being chaos and becomes simple arithmetic.

Now watch what happens with small numbers. Locker 1 has only one divisor, student 1, so it gets opened once and stays open. Locker 2 has two divisors, students 1 and 2. The first opens it, the second closes it, and it ends shut. Locker 3 follows the same pattern: opened by 1, closed by 3, finished closed.

Locker 4 tells a different story. Its divisors are 1, 2, and 4. That is three touches. Open, close, open. It survives the night wide open. The pattern snaps into focus: when the divisor count is even, the locker ends closed. When it is odd, the locker ends open.

So which numbers have an odd number of factors? Most numbers pair their factors neatly. The factors of 24 line up as (1, 24), (2, 12), (3, 8), and (4, 6). Four pairs, eight factors, even count. But a perfect square breaks the pairing. Consider 16: its factors are (1, 16), (2, 8), and (4, 4). That middle pair collapses into a single factor because 4 times 4 uses the same number twice. Three factors, odd count, locker left open.

That is the hidden signature the principal finds the next morning. The only lockers standing open are the perfect squares from 1 to 100: 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. Ten lockers, all with that odd factor count, all quietly announcing the mathematical prank that unfolded overnight.

Why Perfect Squares Win the Locker Puzzle

The ten lockers left swinging open belong to the perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. That tidy list is the entire answer to the locker puzzle, and it all comes down to counting factors. Once you see why factors decide the fate of each door, the pattern clicks into place.

Think about locker number 24. It gets touched by students 1, 2, 3, 4, 6, 8, 12, and 24, exactly the numbers that divide 24 evenly. That is eight touches total, an even count. Since the first student opens it and every touch after that flips its state, an even number of touches means the door ends up closed. Most numbers behave this way because their factors naturally pair up. For 24, the pairs stack neatly as (1, 24), (2, 12), (3, 8), and (4, 6).

But some numbers refuse to pair cleanly. Take 16. Its factors are 1, 2, 4, 8, and 16. The middle factor, 4, pairs with itself, so it only gets counted once. That leaves 16 with five factors, an odd count, which means an odd number of touches. The door gets flipped an odd number of times and finishes open. Numbers with this self-pairing middle factor are perfect squares, and they are the only ones that pull this trick off.

So when the principal walks in the next morning, those ten doors stand open while the other ninety stay shut. The pranksters had no idea their game would line up so neatly with square numbers, but that is exactly what the math promises.