The Checkerboard Problem

I began by finding out how many 1 x 1 squares are on the board I began by finding out how many 1 x 1 squares are on the board. First, I counted the number of squares that is in a row and multiplied by the number of that same size square that is in a column. Basically, I’m finding the area of the whole checkerboard. Thus, for the 1x1 squares I am multiplying 8 times 8 or 82 which is 64 total squares.

Now the two by two squares (4 sq Now the two by two squares (4 sq. units) have to be approached a bit differently. Since each square takes up two squares in a row, you can’t count the last square in the row. If you did, you would have a 1 x 2 rectangle. Because you can’t count the last square in the row, that same strategy would apply to the column which leaves you with multiplying 7 x 7 which is 49.

Thus, there are 49 squares that have an area of 4 square units on a checkerboard. 1 2 3 4 5 6 7 1 2 3 7 x 7 = 7² = 49 4 5 6 7

I noticed a pattern. When I worked with the 3 x 3 squares, I didn’t need to count the last two columns or rows, so you are saying 6 x 6 or 6² which is 36. When I worked with the 2 x 2 squares, I didn’t count the last row or column, so you are saying 7 x 7 or 7² which equals 49.

What I’m doing is counting how many squares of a certain size (n) can fit into the top row; then I’m squaring that number (n²) to find the amount of squares that size can fit into the checkerboard. Also, I noticed that if you take one side of the 2 x 2 square or any other size, you can subtract that one side from 9 and get how many squares will fit in the top row. Then you multiply it by itself to get the total amount.

(9 – n)² = the number of a specific size square. For example: (9 –2)² or 7² is 49, the number of 2 x 2 squares that are on the checkerboard. It works for any size square: (9 – n)² = the number of a specific size square. Now that we have the pattern, we can solve for all sizes of squares.

Size of Square Expression Quantity of Squares 1 x 1 (9 - 1)² 8² = 64 2 x 2 (9 - 2)² 7² = 49 3 x 3 (9 - 3)² 6² = 36 4 x 4 (9 -4)² 5² = 25 5 x 5 (9 - 5)² 4² = 16 6 x 6 (9 - 6)² 3² = 9 7 x 7 (9 - 7)² 2² = 4 8 x 8 (9 - 8)² 1² = 1

64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 Finally, you add all of your totals together: 64 + 49 + 36 + 25 + 16 + 9 + 4 + 1 And find that there are exactly 204 “perfect squares” on a checkerboard!