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How many unique squares are there in a 4 x 4 grid?

Ask the students how many squares are there in a 4x4 grid of squares. Many will answer 16. Tell them that there are more than that.

After they have had a chance to think about and have yelled out some more answers ask them how many squares there are in a 1x1 grid (1) and in a 2x2 grid (the 4 small squares and the 1 big square = 5) and a 3x3 grid (9 small squares, 4 of the 2x2, and 1 big one = 14).

So the total for a 4x4 is 16 + 9 + 4 + 1 = 30. The total for any nxn square is (n squared + n-1 squared + n-2 sqaured down to 1 when n=1.

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