Square problem
The free think problem we were given looks simple and easy and it can be. The problem itself is simply how many squares in a square that looks like this:
When you first look at the problem it only looks like one square but really there are squares within this square. The goal in solving this problem is to count all the squares both big and small.
What we did to solve the problem was very simple at first. We started by simply counting the squares after a couple of minutes of counting we finally got our exact answer. Then we were told to make an equation for the problem and after a little bit of tweaking and fiddling we got our equation to the problem. After that we wanted to find out how many rectangles were in this square. That itself took longer to count and after a couple minutes and a little bit of rechecking our answers we finally got our answer. Unfortunately we didn't get an equation for it yet.
The solution we came up with to the problem was to count all the squares within the square. The number we got was 30 squares. When we counted the squares we only counted ones going the shortest distance between dots. We counted four different types of squares 4x4, 3x3, 2x2, and 1x1. When it comes to an equation our group found one that solves the problem.
What I learned from this problem was how to solve a problem with multiple layers in it. If I were to give myself a grade for this assignment I would give myself a B+. The reason I believe this grade is fair is because although I didn't know how to solve this problem originally I have a fair idea on how to solve the problem in the future. I still will count the squares individually in order to make sure I got the right solution.
When you first look at the problem it only looks like one square but really there are squares within this square. The goal in solving this problem is to count all the squares both big and small.
What we did to solve the problem was very simple at first. We started by simply counting the squares after a couple of minutes of counting we finally got our exact answer. Then we were told to make an equation for the problem and after a little bit of tweaking and fiddling we got our equation to the problem. After that we wanted to find out how many rectangles were in this square. That itself took longer to count and after a couple minutes and a little bit of rechecking our answers we finally got our answer. Unfortunately we didn't get an equation for it yet.
The solution we came up with to the problem was to count all the squares within the square. The number we got was 30 squares. When we counted the squares we only counted ones going the shortest distance between dots. We counted four different types of squares 4x4, 3x3, 2x2, and 1x1. When it comes to an equation our group found one that solves the problem.
What I learned from this problem was how to solve a problem with multiple layers in it. If I were to give myself a grade for this assignment I would give myself a B+. The reason I believe this grade is fair is because although I didn't know how to solve this problem originally I have a fair idea on how to solve the problem in the future. I still will count the squares individually in order to make sure I got the right solution.
Locker problem
The problem I did for Free-Think Friday was called the Locker Problem. To solve the initial problem you needed to find out what lockers were left open and why. The first person opens all 552 lockers, the second person changes every other lockers state, the third person would change every third locker, and so on till the 552 person changes the 552 locker.
To solve this problem my group checked what lockers were open and closed to the 25th locker. We started to see a pattern, but were running out of room in our notebooks. So we started to use a google spreadsheet to check the status of the lockers at a certain person. By doing this we could go to larger numbers and clearly see whether the locker was open or closed.
We found the solution to the main problem relatively quickly. After checking and double checking the solution we found out that the only lockers open were lockers on perfect squares, as you can see in the picture. We also found out that numbers that were touched twice were prime numbers. The whole idea around the locker problem are factors. Lockers touched 3 times had only 3 factors and there were more than one lockers touched the most, 24 times.
What I learned from this problem was that sometimes it is useful to just write out an entire problem rather than doing it in your head. If I had to rate myself on this problem I would definitely give myself a 10 out of 10. I believe I deserve this grade because I not only fully understood how to sole the problem, but I also worked with my group very well.
To solve this problem my group checked what lockers were open and closed to the 25th locker. We started to see a pattern, but were running out of room in our notebooks. So we started to use a google spreadsheet to check the status of the lockers at a certain person. By doing this we could go to larger numbers and clearly see whether the locker was open or closed.
We found the solution to the main problem relatively quickly. After checking and double checking the solution we found out that the only lockers open were lockers on perfect squares, as you can see in the picture. We also found out that numbers that were touched twice were prime numbers. The whole idea around the locker problem are factors. Lockers touched 3 times had only 3 factors and there were more than one lockers touched the most, 24 times.
What I learned from this problem was that sometimes it is useful to just write out an entire problem rather than doing it in your head. If I had to rate myself on this problem I would definitely give myself a 10 out of 10. I believe I deserve this grade because I not only fully understood how to sole the problem, but I also worked with my group very well.