How many squares are there in the given figure?

Concept

In a regular n by n grid, a square of side k cells can start in n - k + 1 horizontal positions and n - k + 1 vertical positions.

Therefore, the number of k by k squares is (n - k + 1)2, and the total is the sum of these counts for k from 1 to n.

Application

1. The figure is a 4 by 4 grid, so n = 4.

2. For 1 by 1 squares, the count is 42 = 16.

3. For 2 by 2 squares, the count is 32 = 9.

4. For 3 by 3 squares, the count is 22 = 4.

5. For 4 by 4 squares, the count is 12 = 1.

6. Adding the counts gives 16 + 9 + 4 + 1 = 30.

Cross-check

The sum-of-squares formula gives n(n + 1)(2n + 1) / 6 = 4 × 5 × 9 / 6 = 30, confirming the direct count.

Thus, the figure contains 30 squares.