Question

How many squares appear in the image given below? 

Hint:

When counting shapes in a complex figure, always categorize them first (e.g., by size, orientation, or type). Count each category separately and then sum the results. This structured approach prevents errors.

Answer 1

Correct Answer: 25

Step 1: Understanding the Concept:
The question requires counting all the squares in a complex geometric figure. The figure contains squares of different sizes and orientations (axis-aligned and rotated).
Step 2: Key Formula or Approach:
A systematic approach is crucial to avoid missing any squares or double-counting. We can categorize the squares by their orientation and then by their size.
Step 3: Detailed Explanation:
Let's divide the squares into two groups: axis-aligned (sides are horizontal and vertical) and rotated.
Group 1: Axis-Aligned Squares

Largest size: The single outermost square. (1)
Medium-large size: Four squares, one in each quadrant, whose corners meet at the center. (4)
Medium size: One square in the center of the figure. (1)
Small size: Four small squares, one near each corner of the outermost square. (4)
Smallest size: One very small square at the exact center of the figure, formed by intersections. (1)
Total axis-aligned squares = \(1 + 4 + 1 + 4 + 1 = 11\).
Group 2: Rotated Squares (tilted at 45 degrees)

Largest size: One large square whose vertices touch the midpoints of the sides of the outermost square. (1)
Medium size: Four squares, each located in one of the 'blades' of the largest rotated square. (4)
Small size: Four squares located at the corners of the central axis-aligned square. (4)
Smallest size: Four small squares, with their vertices touching the corners of the largest rotated square. (4)
Tiny size: One very small square at the exact center, inside the smallest axis-aligned square. (1)
Total rotated squares = \(1 + 4 + 4 + 4 + 1 = 14\).
Total Count:
Total number of squares = (Total axis-aligned squares) + (Total rotated squares)
\[ \text{Total Squares} = 11 + 14 = 25 \] Step 4: Final Answer:
By systematically counting all squares of different sizes and orientations, we find there are a total of 25 squares in the image.