In a school, pupils were told to design a series of patterns with triangles and squares. Riordan's group came up with the following series of patterns.

1. How many squares did Riordan's group need to form pattern 43?
2. How many triangles did Riordan's group need to form a pattern with 33 squares?

(a)
Patterns for the number of triangles and squares
Pattern 1: 1 triangle 5 squares
Pattern 2: 2 triangles 9 squares
Pattern 3: 3 triangles 13 squares

Formula:
Number of triangles = Pattern number

Pattern for the number of squares
Pattern 1: 1 + 1 x 4 = 5
Pattern 2: 1 + 2 x 4 = 9
Pattern 3: 1 + 3 x 4 = 13

Formula:
Number of squares = 1 + Pattern number x 4

Number of squares needed to form Pattern 43
= 1 + 43 x 4
= 173

(b)

1 + Pattern number x 4 = 33
Pattern number x 4 = 33 - 1 = 32
Pattern number = 32 ÷ 4 = 8

Pattern number = Number of triangles
Pattern number 8 = 8 triangles

Answer(s): (a) 173; (b) 8