I first misunderstood the question and thought there were only three students taking turns. Because of that, I started with the first 15 lockers and worked out what would happen to each one. When I read the question again, I realized that there were actually 1,000 students. I decided to increase my example to 30 lockers and made an Excel sheet so I could keep track of what was happening and hopefully see a pattern.
While working through the lockers, I noticed that a student only changes a locker when their student number is a factor of that locker number. For example, locker 12 is changed by students 1, 2, 3, 4, 6, and 12. That means it changes six times and ends up open again. Locker 16 is changed by students 1, 2, 4, 8, and 16, so it changes five times and ends up closed.
This is where the pattern started to make sense to me. Most numbers have factors that come in pairs, so those lockers are changed an even number of times and end up open. Perfect squares are different because they have an odd number of factors. For example, with 16, one of the factor pairs is , so 4 is only counted once.
The perfect squares go 1, 4, 9, 16, 25, ... all the way to 961. Since and , there are 31 perfect squares between 1 and 1,000. This means 31 lockers will be closed and 969 lockers will be open.
My Mathematical Thinking
For this problem, I started with a smaller example, organized my work in Excel, and looked for patterns. I also tested different lockers to see if the pattern continued. Even though I misunderstood the question at first, working through that mistake actually helped me understand the problem better. Once I saw the connection between the locker numbers and their factors, I was able to use that pattern to figure out what would happen with all 1,000 lockers.
No comments:
Post a Comment