For a simple $n \times n$ grid, the total number of squares is given by the sum of squares: $1^2 + 2^2 + \dots + n^2$. For modified grids, count the base grid first and then add the individual extra squares created by the modification.
Step 1: Understanding the Concept:
Counting squares in a complex figure requires a systematic approach, identifying base grids and additional smaller or larger squares formed by overlapping lines. Step 2: Detailed Explanation:
1. First, identify the basic $3 \times 3$ grid structure. A standard $3 \times 3$ grid contains:
- $3 \times 3 = 9$ small ($1 \times 1$) squares.
- $2 \times 2 = 4$ medium ($2 \times 2$) squares.
- $1 \times 1 = 1$ large ($3 \times 3$) square.
Total in basic grid = $9 + 4 + 1 = 14$ squares.
2. Now, observe the secondary grid or the lines added inside. In this specific diagram, there are additional small squares formed in the central regions due to the overlapping structure. There are 9 additional small squares visible within the interior overlapping region.
3. Total squares = (Squares in primary $3 \times 3$ grid) + (Additional interior small squares).
Total = $14 + 9 = 23$. Step 3: Final Answer:
The total number of squares in the figure is 23.
