POW #1: Checkerboard Squares
A problem involving both patterns and and logic, the Checkerboard Squares assignment helped push my cognitive thinking to another level. By learning to recognize similarities and common traits that certain numbers shared, I was able to find and apply a pattern to all aspects of the problem, checking to see if it fit. Being able to locate a pattern is essential to many questions in math, which is why this assignment was so beneficial to my understanding of delving deeper into a problem.
Part 1: Problem Statement
In the Checkerboard Squares problem, the main objective was to find the total number of squares in an 8-by-8 checkered grid. These squares could be any size, (e.g., 1 square by 1 square or 6 squares by 6 squares,) as long as each square was within the boundaries of the 8-by-8 checkerboard. In fact, squares could overlap, meaning that any number of tiles of one square could also be a part of another square, as long as it was not the exact same one. By using the grid we were given, we had to find all squares of sizes ranging from 1-by-1 to 8-by-8, making the task more complex than just counting each square on the checkerboard.
Part 2: Process
Looking at the problem as a whole, I have to admit I was a little overwhelmed at first. (How can I find every single square on this grid without being incorrect or going insane?) In order to help alleviate the stress I was feeling about the upcoming task, I decided to break it down. I knew I had to find squares of every size from 1-by-1 to 8-by-8, so I set up a table:
Part 1: Problem Statement
In the Checkerboard Squares problem, the main objective was to find the total number of squares in an 8-by-8 checkered grid. These squares could be any size, (e.g., 1 square by 1 square or 6 squares by 6 squares,) as long as each square was within the boundaries of the 8-by-8 checkerboard. In fact, squares could overlap, meaning that any number of tiles of one square could also be a part of another square, as long as it was not the exact same one. By using the grid we were given, we had to find all squares of sizes ranging from 1-by-1 to 8-by-8, making the task more complex than just counting each square on the checkerboard.
Part 2: Process
Looking at the problem as a whole, I have to admit I was a little overwhelmed at first. (How can I find every single square on this grid without being incorrect or going insane?) In order to help alleviate the stress I was feeling about the upcoming task, I decided to break it down. I knew I had to find squares of every size from 1-by-1 to 8-by-8, so I set up a table:
Since it was a given that there were 64 1-by-1 squares on the 8-by-8 checkerboard, I filled in both the first and last boxes. Now it was a matter of determining how many squares were in the other six categories. I wanted to start with the number of 7-by-7 squares, since I knew that the larger the square, the easier it would be to find, since there would be less of them. Tracing each possibility of 7-by-7 squares on the checkerboard, my paper looked like this (see right):
In total, I counted four different 7-by-7 squares, since we were allowed to overlap certain tiles. I used the borders to help me determine that there were four squares of this size. This data helped me fill in another section of the table, which now looked like this: |
It was here that I started to notice a pattern. I saw that there were 64 1-by-1 squares, and there was one 8-by-8 square. Knowing that 1 x 1 = 1 and 8 x 8 = 64, there was a similarity in the two sizes of squares. If I flip-flopped the answers, it would be the square size of 1 x 1 = 64 squares, and the size of 8 x 8 = 1 square. Using this logical reasoning, I also knew that 2 x 2 = 4 and 7 x 7 = 49, and there were four 7-by-7 squares, which would make the square size of
7 x 7 = 4 squares. Would the square size of 2 x 2 = 49 squares? Once again, I traced out all of the possibilities for a square size, this time counting the number of 2 x 2 squares in the checkerboard (see left).
Sure enough, there were forty-nine 2-by-2 squares, which all but confirmed my theory about opposite numbers matching. With two pairs of squares with numbers matching each other, I knew I was on the right track, and it would be easy for me to complete the table. After checking the number of 3-by-3 squares and seeing that there were 36 of them, I could finish filling in the empty sections. My table now looked like this:
7 x 7 = 4 squares. Would the square size of 2 x 2 = 49 squares? Once again, I traced out all of the possibilities for a square size, this time counting the number of 2 x 2 squares in the checkerboard (see left).
Sure enough, there were forty-nine 2-by-2 squares, which all but confirmed my theory about opposite numbers matching. With two pairs of squares with numbers matching each other, I knew I was on the right track, and it would be easy for me to complete the table. After checking the number of 3-by-3 squares and seeing that there were 36 of them, I could finish filling in the empty sections. My table now looked like this:
Part 3: Solution
The numbers were all in, and I was successful in solving the problem. However, my work wasn’t completely done. The last step of the problem was to find the number of total squares on the checkerboard, so I added all of the sums of squares to get the final answer.
The numbers were all in, and I was successful in solving the problem. However, my work wasn’t completely done. The last step of the problem was to find the number of total squares on the checkerboard, so I added all of the sums of squares to get the final answer.
But I needed another way to prove my results; a way to connect all of the numbers. I had made connections between pairs of numbers, (e.g., 1 and 8, or 2 and 7) but I needed a way to find a pattern that linked all of the square sizes together. I took a look at the digits in the “Number of Squares” column, and that’s where I found my proof.
Starting at 64, the difference between each number decreased by 2 every time:
Starting at 64, the difference between each number decreased by 2 every time:
While my method may have been unorthodox, it certainly worked. By breaking the information down into a size-by-size table, I was able to take the problem one step at a time.