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Oct. 3, 2007
Checkerboard Squares
A checkerboard is a square that has been divided up into 64 smaller squares by straight lines at right angles. How many squares are there on a checkerboard? The answer is not 64 or 65.
Bonus:
How could you work out how many squares there are on any sized square board? Is there a formula or a procedure for finding the answer?
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Start with a smaller board that is three squares by three squares. You'll see it has nine small squares, four different overlapping larger squares, and the outside square, for a total of 14. Then try a four-by-four board and, step by step, work up to an eight-by-eight checkerboard.
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There are 204 squares on an eight-by-eight checkerboard.
A board that is one square wide has just 1 square.
A two-by-two board has 1 large square and 4 small squares.
A three-by-three board has 1 large square, 4 two-by-two squares, and 9 small squares.
A four-by-four board has 1 large square, 4 three-by-three squares, 9 two-by-two squares, and 16 small squares.
A five-by-five board has 1 large square, 4 four-by-four squares, 9 three-by-three squares, 16 two-by-two squares, and 25 small squares.
See the pattern?
An eight-by-eight board has 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 squares, a total of 204.
Suppose you have a board that is n squares wide. In that case, the total number of squares would be 1 + 4 + 9 + . . . + n2<.
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