Hello, I'm an expert in combinatorial mathematics and problem-solving. Let's delve into the problem of finding out how many squares can be made from a 4x4 grid.

To begin with, we need to understand what constitutes a square in a grid. In a grid, a square can be formed by connecting four points that are aligned either horizontally, vertically, or diagonally. The size of the square can vary; it can be a 1x1 square, a 2x2 square, and so on.

Now, let's break down the problem systematically:


1. 1x1 Squares: These are the smallest squares possible, and they are formed by each individual cell of the grid. Since we have a 4x4 grid, there are \(4 \times 4 = 16\) of these squares.


2. 2x2 Squares: These squares are formed by grouping four 1x1 squares together. To find the number of 2x2 squares, we consider the number of ways we can choose two rows and two columns. There are 3 ways to choose the first row (since there are 4 rows and we cannot choose the last one without choosing the first), and similarly, 3 ways to choose the second row. The same logic applies to choosing two columns. This gives us \(3 \times 3 = 9\) ways to form 2x2 squares.


3. 3x3 Squares: For a 3x3 square, we are essentially choosing three rows and three columns. There are \(4 - 3 + 1 = 2\) ways to choose the first row, and the same for the second and third rows. The same logic applies to columns. This gives us \(2 \times 2 = 4\) ways to form 3x3 squares.


4. 4x4 Squares: There is only one way to form a 4x4 square, which is by taking the entire grid.

Adding these up, we get the total number of squares:
\[ 16 (\text{1x1 squares}) + 9 (\text{2x2 squares}) + 4 (\text{3x3 squares}) + 1 (\text{4x4 square}) = 30 \]

This is the number of squares that can be formed within a 4x4 grid. It's important to note that this calculation takes into account all possible squares without overlap, ensuring that each square is counted once.

Now, let's move on to the next step.

read more >>

After they have had a chance to think about and have yelled out some more answers ask them how many squares there are in a 1x1 grid (1) and in a 2x2 grid (the 4 small squares and the 1 big square = 5) and a 3x3 grid (9 small squares, 4 of the 2x2, and 1 big one = 14). So the total for a 4x4 is 16 + 9 + 4 + 1 = 30.May 18, 2011read more >>