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This problem and the one on the next page are typical of those that students in the Interactive Mathematics Program encounter each day.

Checkerboard Squares

The Question:

How many squares are there in a regular 8-square-by-8-square checkerboard? (The answer is not 64.)

The Solution:

First-year IMP students typically solve this problem by looking for patterns. They can quickly get the number of 1-by-1 squares by multiplying 8 times 8, which is 8 squared. They then count squares made up of other squares. The number of 2-by-2 squares adds up to 49, or 7 squared. The number of 3-by-3 squares equals 36, or 6 squared. A pattern emerges. Each time the dimension of the square is increased, the number squared to get the total decreases by one. By adding these totals together, students get their answer: 204. There is a formula for this sum of squares, which looks like this:

äx2 = 12 + 22 + 32 + 42+ 52 + 62 + 72 + 82 = 204

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