In this sequence of patterns, the number of square tiles is increasing as the pattern number increases. Based on the image, we can determine the following:

• Pattern 1 has 1 square tile.
• Pattern 2 has 3 square tiles.
• Pattern 3 has 6 square tiles.

This suggests a pattern in the number of square tiles, likely related to triangular numbers, where each subsequent term is the sum of the previous number plus an increasing number.

The first few triangular numbers are:

• $T_1 = 1$
• $T_2 = 3$
• $T_3 = 6$
• $T_4 = 10$
• $T_5 = 15$, and so on.

The general formula for the $n$-th triangular number is given by:

$T_n = \frac{n(n+1)}{2}$

We are asked to find the number of square tiles in the 12th pattern. Using the formula:

$T_{12} = \frac{12(12+1)}{2} = \frac{12 \times 13}{2} = 78$

So, there are 78 square tiles in the 12th pattern.

Would you like further details or have any questions?

Here are 5 related questions for deeper exploration:

1. How many triangular tiles are in the 12th pattern?
2. Can you derive a general formula for the total number of tiles (square + triangular) in the $n$-th pattern?
3. How many total tiles (square and triangular) are in the 10th pattern?
4. How does the number of triangular tiles grow compared to the number of square tiles?
5. What is the sum of the square tiles from the 1st to the 12th pattern?

Tip: Recognizing sequences, especially triangular or square numbers, helps in quickly identifying patterns and applying formulas.