POW #1
The Problem:
During the week of September 12 we worked on the Checkered Square Problem. We had to figure out how many 1 x 1, 2 x 2, 3 x 3, etc squares could fit inside an 8 x 8 checkerboard. The squares inside the checker board could overlap but you could not draw the same square twice. No square could overlap onto outside of the checkerboard range. Every shape had to be a perfect square. No rectangles, no parallelograms, only squares.
The Solution:
The solution was very very hard to find. At first I thought I had it, I thought I hit the nail on the head. I drew out the 8 x 8 checkerboard on a piece of graph paper and drew four 3 x 3 squares inside. I had the solution. Four 4 x 4, Four 3 x 3, sixteen 2 x 2. Then a student in the classroom asked if the squares could overlap. The answer was in fact “Yes.” My answer meant nothing now so I had to start back from square (get it) one. At least one thing I did was correct, there were sixty-four 1 x 1 squares. That was a start. Then I said to myself, “There must be sixty-four 2 x 1 squares if they can overlap.” I tested it out and there were in fact sixty-four 2 x 1 squares. Although the solution needed to be perfect squares it was a start. I then tried to visually map out each 3 x 3 square on an 8 x 8 grid. The visual got extremely cluttered and unorganized. I got the answer seventeen 3 x 3 squares. No one in the class came up with the same answer. I knew something was off and that trying to visualize it was definitely not a good choice. Visualization cannot always be the best way to solve. I was honestly completely stumped. I tried to visualize 2 x 2 as well, but to no surprise I ended up with around 30, a number fairly low compared to others. Starting small wasn’t working so I started with the second largest square possible, 7 x 7. 7 x 7 squares could only overlap so much, so I tested it out. I visualized it and actually came up with a pretty solid answer. Four 7 x 7 squares was what I came up with, I wasn’t the only one with this solution. Here I was starting at the top. I knew there were sixty-four 1 x 1 squares, four 7 x 7 squares, and one 8 x 8 square. I tried to visualize 6 x 6 but I did not work whatsoever. I was completely stuck once again. From here I started inputting random rectangles to try to get some sort of result. I didn’t really get anywhere and was just doing random work. Eventually I had to ask for some help. I asked a classmate his approach to the problem and he described that you have to visualize the problem as the square pushing upward. I tried this for each square size and the result was sixty-four 1 x 1, forty-nine 2 x 2, thirty-six 3 x 3, twenty-five 4 x 4, sixteen 5 x 5, nine 6 x 6, four 7 x 7, and one 8 x 8. I checked my answers with others and they were relatively similar
Further Explanation:
During the week of September 12 we worked on the Checkered Square Problem. We had to figure out how many 1 x 1, 2 x 2, 3 x 3, etc squares could fit inside an 8 x 8 checkerboard. The squares inside the checker board could overlap but you could not draw the same square twice. No square could overlap onto outside of the checkerboard range. Every shape had to be a perfect square. No rectangles, no parallelograms, only squares.
The Solution:
The solution was very very hard to find. At first I thought I had it, I thought I hit the nail on the head. I drew out the 8 x 8 checkerboard on a piece of graph paper and drew four 3 x 3 squares inside. I had the solution. Four 4 x 4, Four 3 x 3, sixteen 2 x 2. Then a student in the classroom asked if the squares could overlap. The answer was in fact “Yes.” My answer meant nothing now so I had to start back from square (get it) one. At least one thing I did was correct, there were sixty-four 1 x 1 squares. That was a start. Then I said to myself, “There must be sixty-four 2 x 1 squares if they can overlap.” I tested it out and there were in fact sixty-four 2 x 1 squares. Although the solution needed to be perfect squares it was a start. I then tried to visually map out each 3 x 3 square on an 8 x 8 grid. The visual got extremely cluttered and unorganized. I got the answer seventeen 3 x 3 squares. No one in the class came up with the same answer. I knew something was off and that trying to visualize it was definitely not a good choice. Visualization cannot always be the best way to solve. I was honestly completely stumped. I tried to visualize 2 x 2 as well, but to no surprise I ended up with around 30, a number fairly low compared to others. Starting small wasn’t working so I started with the second largest square possible, 7 x 7. 7 x 7 squares could only overlap so much, so I tested it out. I visualized it and actually came up with a pretty solid answer. Four 7 x 7 squares was what I came up with, I wasn’t the only one with this solution. Here I was starting at the top. I knew there were sixty-four 1 x 1 squares, four 7 x 7 squares, and one 8 x 8 square. I tried to visualize 6 x 6 but I did not work whatsoever. I was completely stuck once again. From here I started inputting random rectangles to try to get some sort of result. I didn’t really get anywhere and was just doing random work. Eventually I had to ask for some help. I asked a classmate his approach to the problem and he described that you have to visualize the problem as the square pushing upward. I tried this for each square size and the result was sixty-four 1 x 1, forty-nine 2 x 2, thirty-six 3 x 3, twenty-five 4 x 4, sixteen 5 x 5, nine 6 x 6, four 7 x 7, and one 8 x 8. I checked my answers with others and they were relatively similar
Further Explanation:
Reflection:
There were many challenges going into this problem. It was very confusing and very hard to visualize. Eventually things worked out and the problem became much clearer. I definitely over complicated the problem when starting out but as soon as I simplified it, it became clear. I tried to used the habits of a mathematician and really focused on starting small, despite it not working. I also focused on keeping things organized, as I always try to do. Overall this problem was not my favorite. It wasn’t a bad problem, I definitely learned something, I just prefer function work. I like to find patterns and creating equations. Visualizing was very necessary in this problem but it got extremely tedious.
There were many challenges going into this problem. It was very confusing and very hard to visualize. Eventually things worked out and the problem became much clearer. I definitely over complicated the problem when starting out but as soon as I simplified it, it became clear. I tried to used the habits of a mathematician and really focused on starting small, despite it not working. I also focused on keeping things organized, as I always try to do. Overall this problem was not my favorite. It wasn’t a bad problem, I definitely learned something, I just prefer function work. I like to find patterns and creating equations. Visualizing was very necessary in this problem but it got extremely tedious.