Question

We can divide a large square into an n x n "checkerboard" of smaller squares. How many squares of any size do the lines of the n x n checkerboard form? Find and prove a formula for this.

Answer 1

The total number of squares formed by the lines of an n x n checkerboard can be found using the sum of squares formula, which is n(n+1)(2n+1)/6.

Step-by-step explanation:

The number of squares formed by the lines of an n x n checkerboard can be found by considering the different possible sizes of squares. We can start with the smallest square, which has a side length of 1, and count the number of squares of each size up to the largest square with a side length of n. For example, for a 3 x 3 checkerboard, we have 9 squares of size 1, 4 squares of size 2, and 1 square of size 3. This pattern continues for larger checkerboards. To find a formula for the total number of squares, we can sum up the number of squares of each size.

Formula: The total number of squares formed by the lines of an n x n checkerboard is given by the sum of the squares of the first n natural numbers, which can be expressed as n(n+1)(2n+1)/6.