A figure consists of a square with 4 equal horizontal and 4 equal vertical lines dividing it into a 5x5 grid. How many squares can be counted in total?

Question

Prepp Team · Accepted Answer

Grid Understanding for Square Counting

The problem asks us to find the total number of squares within a larger square which is divided into a 5x5 grid. This grid structure is created using 4 horizontal and 4 vertical lines inside the square.

Method for Counting Squares

To find the total number of squares, we need to identify and count squares of all possible sizes within the grid. For a grid of size \(n \times n\), the possible square sizes range from \(1 \times 1\) up to \(n \times n\). The general formula for the total number of squares in an \(n \times n\) grid is the sum of the squares of the first \(n\) integers:

\(\text{Total Squares} = \sum_{i=1}^{n} i^2 = 1^2 + 2^2 + 3^2 + \dots + n^2\)

Calculation for the 5x5 Grid

In this case, we have a \(5 \times 5\) grid, so \(n=5\). We need to calculate the sum:

\(\text{Total Squares} = 1^2 + 2^2 + 3^2 + 4^2 + 5^2\)

Let's break down the count for each square size:

Square Size	Number of Squares Possible
\(1 \times 1\)	\(5 \times 5 = 25\)
\(2 \times 2\)	\(4 \times 4 = 16\)
\(3 \times 3\)	\(3 \times 3 = 9\)
\(4 \times 4\)	\(2 \times 2 = 4\)
\(5 \times 5\)	\(1 \times 1 = 1\)