POW: Locker Problem
Written By: Nicki Hwang
October, 2003
PROBLEM STATEMENT:
There are 1000 lockers in the long hall of Westfalls High. In preparation for the beginning of the school, the janitor cleans the locker and paints fresh numbers on the locker doors. The lockers are numbered from 1 to 1000. When the 1000 Westfalls High students arrive from summer vacation, they decide to celebrate of school by working off some energy.The first student, student 1, runs down the row of lockers and opens every door.
Student 2 closes the doors of the lockers 2,4,6,8, and so onto the end of the line.
Student 3 changes the state of the doors of the lockers 3,6,9,12, and so on to the end of the line. ( The student opens the door if it is closed and closes the door if it is opened.) Student 4 changes the state of the doors of the lockers 4,8,12,16, and so on.
Student 5 changes the state of every fifth door.
Student 6 changes the state of every sixth door, and so on until all 1000 students have had a turn.
PLAN:
When I first read the problem, I thought It was going to be impossible because from the first POW that we did ourselves I learned that 1 million was bigger then I thought. So, I thought, would a thousand be bigger then I think it is? I think this problem is kind of similar to the first one we did by ourselves because from that POW it made me realize how big a million was. Now that I think of it I wonder if this POW will make me think of how big one thousand is. Mt hypothesis for this problem is 500.WORK:
I first made a chart 30 by 30 (see last page for chart). I made a chart following the description the problem gives you:"The first student, student 1, runs down the row of lockers and opens every door.
Student 2 closes the doors of the lockers 2,4,6,8, and so onto the end of the line.
Student 3 changes the state of the doors of the lockers 3,6,9,12, and so on to the end of the line. ( The student opens the door if it is closed and closes the door if it is opened.) Student 4 changes the state of the doors of the lockers 4,8,12,16, and so on.
Student 5 changes the state of every fifth door.
Student 6 changes the state of every sixth door, and so on until all 1000 students have had a turn"
While finding doing just a 30 by 30 chart, I found that there was a pattern going on, the first locker was always opened. Then I found that number lockers 2 and 3 were not opened then locker number 4 was. I went on with my chart then found out that lockers 5,6,7, and 8 were all opened and locker number 9 was closed. By then I noticed a small pattern, that each time two closed lockers were added on. What I mean is that locker numbers 2 and 3 were opened (two lockers were opened), then lockers 5,6,7, and 8 were opened (4 lockers opened). I went on with my chart and found out that each time two more lockers were added on from they previous number of lockers then one locker was closed. Because it worked for all the first part of my chart (30 by 30), I used my best judgement that I would work for the rest. So, I went on from 25 (the last number that would fit in the 30 by 30 chart) and each time I added two more lockers to the number of lockers I had added before. Basically this:
3+2=5 5+2=7 7+2=9 9+2=11
1+3=4 4+5=9 9+7=16 16+9=25 25+11=36 and so on.
Finally after doing all the equations, I counted up all my equations (the arrows on the bottom point to the kind of equation I counted to get how many lockers were opened).
ANSWER:
My answer was 31. Yes, I do think my answer makes sense. No, I don’t think there could be other answers to this problem. My hypothesis was a lot larger then my final answer. I was 469 lockers off! I think from this problem I learned to make better charts and I think I know how big one thousand is now. I also think that even though that a problem seems complicated you just have to look at it for an hour of so and then, it won’t seem as complicated as you think it really is.