The diagram shows a sequence of patterns formed by identical triangles.

1. Find the values of (i), (ii) and (ii). Give your answers in the following format. (Eg 1, 2, 3)
2. A figure in the pattern has a total of 64 triangles. What is the Figure Number?
3. Another figure has 42 more shaded triangles than unshaded triangles. What is the total number of triangles in this figure?

(a)
Number of shaded triangles:
Figure 1: 1 = 1
Figure 2: 3 = 1 + 2
Figure 3: 6 = 1 + 2 + 3

Formula:
Number of shaded triangles = ¹₂ x Figure Number x (Figure Number + 1)

Number of unshaded triangles:
Figure 1: 0
Figure 2: 1 = 1
Figure 3: 3 = 1 + 2

Formula:
Number of unshaded triangles = ¹₂ x (Figure Number - 1) x (Figure Number)

Total number of triangles:
Figure 1: 1 = 1 x 1
Figure 2: 4 = 2 x 2
Figure 3: 9 = 3 x 3

Formula:
Total number of triangles = Figure Number x Figure Number

Number of shaded triangles in Figure 5
= ¹₂ x Figure Number x (Figure Number + 1)
= ¹₂ x 5 x 6
= 15

Number of unshaded triangles in Figure 5
= ¹₂ x (Figure Number - 1) x (Figure Number)
= ¹₂ x 4 x 5
= 10

Total number of triangles in Figure 5
= 5 x 5
= 25

(b)
Figure Number that has a total of 64 triangles
= √64
= 8

(c)
Number of more shaded triangles than unshaded triangles:
Pattern 1: 1 - 0 = 1
Pattern 2: 3 - 1 = 2
Pattern 3: 6 - 3 = 3

Formula:
Number of more shaded triangles than unshaded triangles = Figure Number

Since there is 42 more shaded triangles than unshaded triangles, it is Figure Number 42.

Total number of squares in Figure Number 42
= 42 x 42
= 1764

Answer(s): (a) 15, 10, 25; (b) 8; (c) 1764