To extend the pattern to Figure 13, we need to analyze the progression of figures in the image:

Observations:

1. Figure 1: Contains $2 \times 2 = 4$ squares, with one additional square space at the bottom right.
2. Figure 2: Contains $3 \times 3 = 9$ squares, with one additional row and column added compared to Figure 1, and an additional square space at the bottom right.
3. Figure 3: Contains $4 \times 4 = 16$ squares, with one more row and column added, and the same additional square space.

Pattern:

Each subsequent figure adds one row and one column, forming a square grid. An additional row and column space are left at the bottom right.

General Formula for $n$-th Figure:

• Total grid squares (without spaces): $n \times n$
• Additional row/column spaces: 1.

Figure 13:

For Figure 13:

• Total grid squares = $13 \times 13 = 169$.
• The bottom-right corner will have an extra space for one row and one column.

Would you like me to provide a diagrammatic representation or more detailed visualization of this?

Related Questions:

1. How can we generalize this pattern mathematically for any figure $n$?
2. What is the total number of empty spaces created up to Figure 13?
3. How does the perimeter of the square evolve as the figure progresses?
4. What real-world applications could such patterns relate to?
5. Can this pattern be extended into three dimensions?

Tip: Look for patterns in row or column growth to simplify calculations in geometric sequences