Problem of the Week- Write Up
Checkerboard Squares
Problem Statement
Checkerboard Squares is the name of our math problem of the week (POW) and we are given a 8-8 checkerboard made of 64 small squares. There are many other dimensions of squares inside this 8-8 checkerboard. For example, 4-4, 3-3, and etc. We are asked to find how many squares are there altogether (including the 64 small squares). You are able to overlap the squares when trying to find a vast amount of your specific sized squares. For the second part of the problem you have to find out if any other sized square besides 8-8 square amount could be determined from a formula equation. You need to find a formula that could easily compute the exact amount of squares inside any large sized square. These two questions were used to find patterns and find formulas that grew your mind set. I made several conjectures during my POW and I couldn't believe how many patterns and squares I found while making several work drafts.
Process Description and Solution
I started with a 7-7 square in a 8-8 square and I learned how to overlap several squares to find new hidden squares. I tried to find any left over smaller squares in the 8-8 square and that definitely helped me find double the squares. I had two rough drafts on the first question because of how many times I took apart and put back together this problem. For example, I tried color coordinated each dimension square in a 8-8 square that I found, but once trying several times and being patient/persistent, I was able to discover more squares. Another example is how many trials I did with making an equation because of how long it takes for me to discover different patterns. For the extensions, I was very patient and persistent to find patterns, more undiscovered squares, and I feel a whole new perspective towards POW’s because of all the habits of a mathematician I learned. I believe that I found exact or close to the amount of squares in a 8-8, 7-7, 6-6, etc sized square. I found this by color coordinating and trying several positions located everywhere on the square. For the second problem I tried several times to make an equation, but I could only find patterns and not a definite formula that works for all square sizes. Fortunately worked on making conjectures and testing them whether or not they work.
Extensions and Further Exploration
I explored trying to find an equation and one thing I noticed was how many ways you could position a square, to find double or treble more squares you spotted in the beginning. I tried several times to find an equation that would instantly calculate the number of squares. But, I could only find patterns and i'm still going to continue with trial and error, until I succeed with this problem. I am happy about trying several drafts to get an exact number of squares in a 8-8 square because of how persistent I was. Also, I know how to break apart a seemingly confusing problem, how to take it apart, and how to find a pattern, and make a strong proven formula. The approach I had for my second draft was to color coordinate and try to find another formula, other than patterns.
Reflection
I found it challenging to find a formula that would work for all square dimensions and continuously tried to find a fitted one. Overall, I explored and discovered patterns I didn't notice at first glance and I will definitely in the future work on staying organized and systematic. I will work on this because now I’ve realized I need to keep track and take notes on everything i’m discovering or thinking of, or else that will be lost. Most importantly, I think some of my strengths I had while working on this POW was conjecture and test because I was very dedicated towards finding a equation and or more patterns. I sought out to accomplish both, but sadly I am still working on a universal equation for all different squares. One question I still have about this problem is how to successfully monitor how many squares you’re finding, but while completing this problem I’ve realized you need to find the basic pattern in the rational numbers in order to find an equation to compute (without self-counting) how many squares total there are.
Checkerboard Squares is the name of our math problem of the week (POW) and we are given a 8-8 checkerboard made of 64 small squares. There are many other dimensions of squares inside this 8-8 checkerboard. For example, 4-4, 3-3, and etc. We are asked to find how many squares are there altogether (including the 64 small squares). You are able to overlap the squares when trying to find a vast amount of your specific sized squares. For the second part of the problem you have to find out if any other sized square besides 8-8 square amount could be determined from a formula equation. You need to find a formula that could easily compute the exact amount of squares inside any large sized square. These two questions were used to find patterns and find formulas that grew your mind set. I made several conjectures during my POW and I couldn't believe how many patterns and squares I found while making several work drafts.
Process Description and Solution
I started with a 7-7 square in a 8-8 square and I learned how to overlap several squares to find new hidden squares. I tried to find any left over smaller squares in the 8-8 square and that definitely helped me find double the squares. I had two rough drafts on the first question because of how many times I took apart and put back together this problem. For example, I tried color coordinated each dimension square in a 8-8 square that I found, but once trying several times and being patient/persistent, I was able to discover more squares. Another example is how many trials I did with making an equation because of how long it takes for me to discover different patterns. For the extensions, I was very patient and persistent to find patterns, more undiscovered squares, and I feel a whole new perspective towards POW’s because of all the habits of a mathematician I learned. I believe that I found exact or close to the amount of squares in a 8-8, 7-7, 6-6, etc sized square. I found this by color coordinating and trying several positions located everywhere on the square. For the second problem I tried several times to make an equation, but I could only find patterns and not a definite formula that works for all square sizes. Fortunately worked on making conjectures and testing them whether or not they work.
Extensions and Further Exploration
I explored trying to find an equation and one thing I noticed was how many ways you could position a square, to find double or treble more squares you spotted in the beginning. I tried several times to find an equation that would instantly calculate the number of squares. But, I could only find patterns and i'm still going to continue with trial and error, until I succeed with this problem. I am happy about trying several drafts to get an exact number of squares in a 8-8 square because of how persistent I was. Also, I know how to break apart a seemingly confusing problem, how to take it apart, and how to find a pattern, and make a strong proven formula. The approach I had for my second draft was to color coordinate and try to find another formula, other than patterns.
Reflection
I found it challenging to find a formula that would work for all square dimensions and continuously tried to find a fitted one. Overall, I explored and discovered patterns I didn't notice at first glance and I will definitely in the future work on staying organized and systematic. I will work on this because now I’ve realized I need to keep track and take notes on everything i’m discovering or thinking of, or else that will be lost. Most importantly, I think some of my strengths I had while working on this POW was conjecture and test because I was very dedicated towards finding a equation and or more patterns. I sought out to accomplish both, but sadly I am still working on a universal equation for all different squares. One question I still have about this problem is how to successfully monitor how many squares you’re finding, but while completing this problem I’ve realized you need to find the basic pattern in the rational numbers in order to find an equation to compute (without self-counting) how many squares total there are.