The 100 Locker Problem

This is probably one of the best math problems I give my fifth grade students. It goes like this:

There are 100 lockers in the long front hall of our school. Each August, the custodians add a fresh coat of paint to the lockers and replace any of the broken number plates. The lockers are numbered from 1 to 100.

When the students arrive on the first day, they decide to celebrate the start of the school year with our school tradition. The first student inside runs down the hall opening all of the lockers. The second student runs down the hall closing every second locker, beginning with locker number 2. The third student reverses the position of ever third locker, beginning with locker number 3. (If the locker is open, she closes it. If it’s closed, she opens it.) The fourth student changes the position of every fourth locker, beginning with number 4. This continues until the 100^th student has a turn, changing the position of the 100^th locker.

At the end of this ritual, which locker doors are open?

Why are the open lockers left open?

Which patterns emerged in your work?

After a week of working on this problem with their partner, they write up what they discovered on posters. I have them include these four sections:

1. Restate the problem

2. Procedure

3. Answer

4. Check

It’s such a fun problem because at first they think it’s impossible, but by the end of the week, all the groups were able to figure it out! We share out our work in a “math congress” where each group presents. Also, students ask comments or give feedback to the groups that present. Here are some of their posters.