Question of the Day
Challenge your mind with the Locker Paradox.
THE LOCKER PARADOX
A school has 100 lockers, numbered 1 to 100, and 100 students.
- Student 1 walks down the hall and opens every locker.
- Student 2 walks down the hall and closes every second locker (lockers 2, 4, 6, 8...).
- Student 3 changes the state of every third locker (lockers 3, 6, 9...). If it is open, they close it; if it is closed, they open it.
This pattern continues until all 100 students have walked past.
Question
After the 100th student walks by, which lockers remain OPEN?
ANSWER
The lockers that remain OPEN are the perfect square numbers:
Total Open Lockers: 10
EXPLANATION
Each locker is toggled once for every factor (divisor) of its locker number.
- A locker toggled an even number of times ends closed.
- A locker toggled an odd number of times ends open.
Most numbers have factors in pairs. For example:
Factors: 1, 2, 3, 4, 6, 12
Total factors: 6 (even) → Closed
Perfect squares have one unpaired factor—their square root. For example:
Factors: 1, 2, 4, 8, 16
Total factors: 5 (odd) → Open
Factors: 1, 5, 25
Total factors: 3 (odd) → Open
Therefore, only lockers numbered with perfect squares remain open after all 100 students have finished.