The locker problem is an open-ended problem where we have 550 lockers set up in the school. There are 550 students who go to school therefore there is one locker for every student.  When the students return from summer vacation they decided to do an experiment where they do a variation of opening and closing lockers. The 1st student opened every single locker from 1 to 550, the 2nd student goes and shuts every other locker starting at 2 to 550, 3rd student changes the state of every 3rd locker ( if it’s open it becomes closed, and vice versa). Then the 4th student comes and changes the state of every 4th locker, the 5th student changes the state of every 5th locker, the 6th student changes the state of every 6th locker, and so on all the way up to 550.
When the students finish, which lockers will be open and which will be closed? Why?
If the students were not in order when they went to open and close lockers with the same lockers be open when they finished? Why?
Which lockers were only touched by 2 students and were any touched by only 3 students?
Which number locker was touched by the most students?

Process description:
When we 1st received the assignment I began to think of ways in which I could solve it such’s doing it with cards, writing about on paper, finding a pattern, doing it algebraically. I decided that I was going to start by doing it with cards in order to find a pattern. At 1st I did not have cards with me so I took a piece of paper colored in one side cut it up into several dozen squares, laid them out  from 1 to 50, and started to turn them over one by one in the specific pattern. Unfortunately, it was very difficult to decipher open from close because of the similarity in shade, and I accidentally lost count and messed up. Then I attempted it one more time this time I filmed by self so that I could keep a record of it. Unfortunately this time I ran out of time and had to stop. After that when I got to my house I got some cards and set up the 1st 50. Then I use my laptop to film myself doing the experiment to keep a record of it. Then I began the pattern. The cards were much easier to use than the paper was because they were easier to decipher open from close, they were easier to turn, and I did not lose count. Well I was turning the cards I noticed a few patterns, but not any patterns to explain how to solve the rest of the problem. Once I finished going through each one and changing it I discovered some very interesting. The cards left open for the 1st 50 were 1, 4, 9, 16, 25, 36, 49. As I looked at these numbers I discovered something. If you subtract 1 from 4 you get 3 then if you had 2 to 3 which equals 5 and you add that to 4 you get 9. Then if you take 5 and add 2 to it you get 7 and then if you 7 to 9 you get 16. How this works is you add 2 to the previous digit that you added 2 to then you add that to the digit that you just came up with and that will give you the next number in the sequence. This was a little bit more complex than it needed to be, and I overlooked something that I should’ve seen. The next day while I was at school I started discussing the locker problem with another student. She had done it quite differently than I did, by writing it out,  so we are discussing how we each did it and what we got. I started explaining the pattern that I found to her, but she didn’t fully understand it until a few days after when I went into more detail about it. At this point I had thought that I had found the perfect solution to the problem. Then in cross we started to talk about the locker problem on the whiteboard, this is what I found out what I had overlooked, we are talking about how we solve the problem when Dr. Drew asked us to look at the problem for a little while and see if we see any other patterns. That’s when I realize all the numbers were perfect squares. 1 equals 1 multiplied by 1, 4 equals 2 multiply by 2, 9 equals 3 multiplied by 3, 16 equals 4 multiplied by 4, and so on. There is a much simpler solution to the one that I had come up with.

Solution:
Out of the 550 lockers what lockers will be open and what lockers will be closed?

Open:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Closed:
2-3, 5-8, 10-15, 17-24, 26-35, 37-48, 50-63, 65-80, 82-99,101-120, 122-143, 145-168, 170-195, 197-224, 226-255, 257-288, 290-323, 325-361, 362-399, 401-440, 442-483, 485-528, 530-550

I know that the solution is complete and correct because you can use any of the 3 methods that I found out about to solve it and it is all the same. If you use the method of turning card you get the same numbers, if you use the method of adding to to the last sum you get the same answers, and if you use perfect square numbers you get the same answer.

If the students were not in the order when they open and close the lockers, would the same lockers be open the end?

Yes, if the students did not go in order when they went to change the state of the lockers the same ones would be open in the same ones would be close. How I found this out was I took the 1st 10 cards, or lockers, and started flipping them going in a random order. When I was done the numbers 1, 4, 9 were still open. From this I made the assumption that the rest of them would follow the pattern no matter the order.

Which lockers were only touched by 2 students, which lockers were only touched by 3 students, and which locker was touched the most?

lockers 2, 3, 5, 7 were only touched by 2 students. They only were touched by 2 students because the first student opened all the and the 2nd student came in close every 2nd one therefore two was only touched twice and 3 was only touched twice because it was first opened by the first student and then the 2nd student skipped it but the 3rd student touched it. 4, 9 were touched by 3 students.

Well doing this problem I have learned more about searching for patterns and how to do it as well as finding alternative patterns even if you have one.  I’ve also learned that you can solve a large problem by doing a small portion of it and expanding from there. If I was to give myself a grade on this open-ended problem it would be 10 out of 10 because I worked very hard on solving this problem, and I was able to do it before we were told how to. I went through a lot of effort experimenting with physical objects and then moving on to a mathematical version as well as I helped out a lot of other students and showed them the way that I solved it and collaborated with of them. The habit of mind that I chose for this open-ended problem is evidence. I used evidence in this problem by not just finding one way to prove that my solution is correct but actually finding 3 ways that all get the same answer, as well as I collaborated with other students to make sure that I was doing it correctly and had similar answers as them.