EDCP 342: Reflection – The Locker Problem

A school has 1000 lockers and 1000 students. On the first morning:

*The first student walks along and opens every locker.

*The second student walks along and closes every second locker.

*The third student walks along and changes the position of every third locker door.

And so on. Once all 1000 students have completed this process, which lockers are open?

Briefly describe how you solved it?

To solve this problem, I first tried reduced the number of lockers to ten to see if there was a pattern that could be found with a smaller set. I manually wrote out how the locker positions would change given ten students walked along and changed the positions for the first ten lockers. After ten students had walked along, the firs ten lockers would o longer be changed. I circled all the locker numbers that were left open and tried to see any pattern of this smaller set. The open lockers were 1, 4, and 9 and these are all the perfect squares found between 1 and 10. I assumed then that all of the perfect squares less than 1000 would be left open once all the students had walked through. From this point, I “guess and checked” the largest number whose square was less than 1000 - this is 31. Thus there are 31 lockers left open after all of the students pass through all 1000 lockers.

Where do you think a student might get stuck?

 A student can be overwhelmed when initially reading the problem. The shear number of lockers and students to account for may present itself as an overly large counting task. If a student does not know how to initially break down the problem into steps that are more approachable then he/she will not feel as if they can find a solution. Even if a student is able to simplify the problem, he/she may not make establish a pattern between the open lockers and perfect squares. Perfect squares are often a concept that takes time to become familiar with recognizing its presence.

How might you assist that stuck student?

To assist students having trouble, especially those intimidated by the problem parameters, can suggest breaking it down into a smaller number of lockers – counting open lockers from 1-20 is more approachable than 1000. When acknowledging patterns, can encourage students to think about how mathematicians describe and classify numbers – whether they are even, odd, prime, and their possible factors.

What extensions could you offer students ready for a challenge?

Those students who are looking to be challenged can be asked to:

·      Find any pattern that exists between the occurrence of perfect squares.

·       Determine how many lockers were open after 100 students pass through? 200 students? 35o students? Is there another pattern found to how may lockers would be left open for any number of students passing through that is less than the number of lockers?