1. How many sticks are needed to form Pattern 12?
2. In which pattern will each side of the triangle measure 64 cm?
3. Calculate the number of shaded triangles in Pattern 67.

(a)
Number of sticks:
Pattern 1: 1 x 3   = (1) x 3 = 3 sticks
Pattern 2: 3 x 3   = (1 + 2) x 3 = 9 sticks
Pattern 3: 6 x 3   = (1 + 2 + 3) x 3 = 18 sticks
Pattern 4: 10 x 3 = (1 + 2 + 3 + 4) x 3 = 30 sticks

Sum of numbers = Pattern Number x (Pattern Number + 1) ÷ 2

Formula:
Number of sticks = Sum of numbers up to Pattern Number x 3

Number of sticks needed for Pattern 12
= (1 + 2 + 3 + ... + 12) x 3
= [12 x (12 + 1) ÷ 2] x 3
= [12 x 13 ÷ 2] x 3
= 78 x 3
= 234

(b)
Length of each side of the triangle:
Pattern 1: 1 x 2 = 2 cm
Pattern 2: 2 x 2 = 4 cm
Pattern 3: 3 x 2 = 6 cm
Pattern 4: 4 x 2 = 8 cm

Formula:
Length of each side of the triangle = Pattern Number x 2

Pattern Number with each side of the triangle measuring 64 cm
= 64 ÷ 2
= 32 

(c)
Pattern 1: Number of shaded triangles = 0
Pattern 2: Number of shaded triangles = 0 + 1
Pattern 3: Number of shaded triangles = 0 + 1 + 2
Pattern 4: Number of shaded triangles = 0 + 1 + 2 + 3

Pattern for number of shaded triangles = Sum of numbers up to (Pattern Number - 1)

Number of shaded triangles in Pattern 67
= [(Pattern Number - 1) x (Pattern Number)] ÷ 2
= (66 x 67) ÷ 2
= 2211

Answer(s): (a) 234; (b) 32; (c) 2211