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
A version of this article appeared in the January 01, 1996 edition of Teacher as Question and Solution