Question and Solution
This problem and the one on the next page are typical of those that students in the Interactive Mathematics Program encounter each day.
How many squares are there in a regular 8-square-by-8-square checkerboard? (The answer is not 64.)
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