Number Theory

UK Primary (3-6)

Locker Problem (Investigation)

Lesson

Once upon a time, there was a most unusual school that had a most unusual first day of school ceremony. In this school there were exactly 100 students and each student had exactly one locker.

On the first day of school all 100 students assembled on the front lawn at exactly 9 AM. When the principal gave the signal, the first student in line entered the school and opened each of the 100 lockers. Then the second student in line entered the school and closed every even numbered locker.

The third student then entered and changed the state of every third locker.

The fourth student then entered the school and changed the state of every fourth locker.

Next, the fifth student then entered the school and changed the state of every fifth locker.

The pattern continued until each of the 100 students have passed through the school and completed his or her locker task. You may assume that each student did each task correctly.

1. After all the students are finished, which lockers are open and which lockers are closed?

2. What if there were 1000 lockers?

3. What do you notice about the lockers that are opened and the lockers that are closed?

4. Why are certain lockers closed and others opened?