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32 33 cuttIng squares a fresh look at an old pattern Beautiful theorems are often arrived at by coming up with new ways of interpreting old patterns. For a whirlwind tour of some classic examples dating back to the Pythagoreans, lets consider various ways of dissecting a square array of n times n, or n2, pebbles. The first way gives the elementary equality n n ... n n times n2. The second way translates into the surprising fact that the sum of the first n odd numbers is equal to n2. For another proof of this theorem, just note that the numbers of triangles in the columns of tiling n below are the first n odd numbers and that after separating the black and the gray triangles lower, opposite, we get a square of n2 triangles. Closely related is the third way of dissecting which corresponds to the equality n 12 2n 1 n2. Choosing the odd number 2n 1 to be a square, we get a Pythagorean triple see page 4. For example, choosing 2n 1 32 gives n 5, and therefore 42 32 52. One last way of dissecting the square array shows that n2 is equal to the sum of the first n natural numbers plus the sum of the first n 1 natural numbers. Can you see how to derive from this a formula for the sum of the first n natural numbers
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