# Locker Problem (Multiples, Squares, Primes, Factors, Patterns)

One hundred students are assigned lockers 1 through 100. The student assigned to locker number 1 opens all 100 lockers. The student assigned to locker number 2 then closes all lockers whose numbers are multiples of 2. The student assigned to locker number 3 changes the status of all lockers whose numbers are multiples of 3 (e.g. locker number 3, which is open gets closed, locker number 6, which is closed, gets opened). The student assigned to locker number 4 changes the status of all locker whose numbers are multiples of 4, and so on for all 100 lockers.

- Which lockers will be left open?

2. Explain how you determined that these particular lockers will be open.

3. What do you notice about these particular locker numbers?

4. Why are these specific locker numbers still open?

5. How many lockers, and which ones, were touched exactly twice? How do you know?

6. Which students touched both lockers 36 and 48? How do you know?

7. Which locker was switched the most times? How do you know?

**K/1st graders**: Question one above is the only applicable questions for grades K/1. The only objective for these grades was for the children to see the patterns in finding multiples or counting numbers. The multiples of one are 1,2,3,4,5 ….. The multiples of 2 are 2,4,6,8,10,12…. They also knew their multiples of 5 and 10 pretty well. Multiples of 3 were more difficult but as they charted out 3,6,9,12…. they saw that there were always two boxes in between each multiple of 3 as there was one box between multiples of 2. Multiples of 4 are fun if you think about them as every other multiple of 2 or every other even number starting with 4.

I also wanted them to see the pattern where each locker number started as open in the first position and then alternated to closed back to open and so forth.

Parents of K/1, just have your children practice their multiples of 1, 2, 3, 4, and 5. I do not expect the children to go beyond this. However, if you wish, go beyond and fill out the attached worksheets for the Locker Problem looking to see which lockers remain open after the last student (100th student) opens locker number 100.

**2/3rd graders**: In addition to the objectives for K/1, I wanted the 2/3rd graders to find the open lockers on their own and to follow the patter of open and closed lockers. Once they were able to find the first four numbers in the pattern: 1, 4, 9, and 16, I asked them to extrapolate (find a pattern) of what would be the next six numbers in the sequence. Once they looked for differences between 1 and 4, 4 and 9, and 9 and 16 and saw +3 +5 +7 they should surmise that the next consecutive odd number difference of +9 +11 … would follow.

As a special challenge, I also created a worksheet which would allow them to graph 39 students and 57 lockers giving them more practice with multiples.

**4-6th grade**: Before we even gave them the worksheet to solve this problem, I challenged them to see how many lockers the first few students had to touch. Student 1 had to touch all 100 lockers; student 2 only 50; student 3 only 33; student 4, only 25; student 5 only 20 and student 6 only 16. Once we spend more time on short division, I will expect them to go beyond this.

In addition to the objectives by the lower grades, I wanted them to be able to identify the name of this amazing sequence as the square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. I spend considerable time focusing them on the WHY. I had the look at the number of times students touched these lockers and they realized that each square number has an odd number of factors so starting with open, closed, open …. it would always end open. We actually factored each of these squares and found they had this number of factors: 1, 3,3,5,3,9,3,7,5,and 9. All of the other lockers had an even number of factors (touched by an even number of students) resulting in a closed locker. Finally, we looked at which lockers were touched exactly twice and they realized that these were the prime numbered lockers which only have two distinct factors (1 and the number itself).

We talked about positive language as we worked through the first 25 primes to 100. When a student said, “15, instead of saying NO, we would simply say 3x5 (the reason why this number is not prime).” We listed all of these numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97.

With two groups, we were even able to talk about which locker numbers were touched the most. There are actually 5 numbers: 60, 72, 84, 90, 96 that each have twelve factors.

Attachment | Size |
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Locker_Problem_K1_27_circles_and_blank.pdf.pdf | 795.85 KB |

Locker_Problem_K1_Answer_Key2.pdf | 118.29 KB |

Locker_Problem_27_and_57_lockers_blank.pdf | 79.85 KB |

Locker_Problem_27_and_57_lockers_Answer_Key.pdf | 107.45 KB |