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Published bySage Jinks Modified about 1 year ago

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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 8 2 which is 64 total squares.

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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.

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Thus, there are 49 squares that have an area of 4 square units on a checkerboard x 7 = 7² = 49

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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.

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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.

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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.

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Size of SquareExpression Quantity of Squares 1 x 1(9 - 1)²8² = 64 2 x 2(9 - 2)²7² = 49 3 x 3(9 - 3)²6² = 36 4 x 4(9 -4)²5² = 25 5 x 5(9 - 5)²4² = 16 6 x 6(9 - 6)²3² = 9 7 x 7(9 - 7)²2² = 4 8 x 8(9 - 8)²1² = 1

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Finally, you add all of your totals together: And find that there are exactly 204 “perfect squares” on a checkerboard!

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