Here are some mathematical problems involving patterns and shapes:

1. Problem: Triangle Number Pattern Consider a sequence of dots arranged in an equilateral triangle. The number of dots increases as follows:

   • 1 dot in the first row
   • 3 dots in the second row
   • 6 dots in the third row
   • 10 dots in the fourth row
   • ...

   a. Find the formula for the number of dots in the nth row of the triangle. b. What is the total number of dots in the 10th row?

2. Problem: Square Tiling Pattern You have a 3x3 square grid where each cell can be either filled or empty. If you fill the cells in the grid such that no two adjacent rows or columns have the same pattern, how many distinct patterns are possible?

3. Problem: Hexagon Growth Pattern Imagine you are adding rings of hexagons around a central hexagon. The first ring consists of 6 hexagons, the second ring consists of 12 hexagons, and the third ring consists of 18 hexagons.

   a. Find the pattern for the number of hexagons in the nth ring. b. How many hexagons will there be in the 5th ring?

4. Problem: Fibonacci Spiral The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, 34, ... These numbers can be represented as squares, where the side length of the square corresponds to the Fibonacci number.

   a. If you arrange these squares next to each other to form a spiral, what is the area of the spiral after adding the first 5 squares? b. What is the ratio of the area of the 5th square to the 4th square?

5. Problem: Star Polygon Angle Pattern A star polygon is formed by connecting every second vertex of a regular polygon. For example, connecting every second vertex of a pentagon forms a star.

   a. If you have a regular 9-sided polygon and connect every third vertex, what type of star is formed? b. What is the internal angle of each point of the star?

---

Tip: Recognizing patterns in shapes often involves both visual and numerical patterns, so always check the geometry as well as the sequence of numbers involved.

Would you like more details on one of these problems, or do you have any questions?

Here are 5 additional questions to consider:

1. How can you generalize the formula for the nth triangular number?
2. What other shapes besides triangles have number patterns associated with them?
3. What is the relationship between Fibonacci numbers and geometry?
4. How can patterns be used to solve tiling problems efficiently?
5. How does symmetry play a role in the creation of star polygons?