Math Problem Statement

How many squares are there in the figure below?

Solution

The image you shared contains a grid of squares. It seems the goal is to calculate how many squares are present within this grid.

To solve this, we must count not just the small squares, but all the larger squares that can be formed within the grid. Here's how you do it:

  1. Smallest squares: These are the individual 1x1 squares in the grid. Since the grid has 6 rows and 6 columns, there are 6 x 6 = 36 such squares.

  2. Larger squares:

    • 2x2 squares: These can be formed by selecting a 2x2 block of smaller squares. The number of 2x2 squares is 5 rows and 5 columns (since one unit is lost on each side), so there are 5 x 5 = 25 of these squares.
    • 3x3 squares: Similarly, there are 4 rows and 4 columns available for a 3x3 square, so there are 4 x 4 = 16 such squares.
    • 4x4 squares: These fit into 3 rows and 3 columns, so there are 3 x 3 = 9 of these squares.
    • 5x5 squares: These fit into 2 rows and 2 columns, so there are 2 x 2 = 4 of these squares.
    • 6x6 square: The entire grid is a single 6x6 square, so there is 1 of these.

Now, we can sum these squares:

  • 1x1 squares: 36
  • 2x2 squares: 25
  • 3x3 squares: 16
  • 4x4 squares: 9
  • 5x5 squares: 4
  • 6x6 square: 1

Total squares = 36 + 25 + 16 + 9 + 4 + 1 = 91 squares.

Would you like further details on this method or any clarifications? Here are some related questions you might find interesting:

  1. How do you calculate the number of squares in different grid sizes?
  2. What is the difference between counting squares and other geometric shapes in grids?
  3. How can this method be generalized to larger or irregular grids?
  4. How does the method change if the grid is non-square (e.g., 4x6)?
  5. Are there other ways to efficiently count squares in a grid?

Tip: To avoid mistakes when counting squares, break the problem down into smaller steps like counting 1x1, 2x2, etc., and avoid skipping over possible combinations.

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Math Problem Analysis

Mathematical Concepts

Combinatorics
Geometry
Counting Strategies

Formulas

Sum of squares formula: n(n+1)(2n+1)/6

Theorems

Counting squares in an n x n grid

Suitable Grade Level

Grades 6-10