A series of patterns of shaded and unshaded small triangles is shown.

1. Which pattern has 87 shaded triangles?
2. Find the number of unshaded triangles in Pattern 41.
3. Find the total number of triangles in Pattern 65.

Pattern	Number of shaded triangles	Number of unshaded triangles	Total number of triangles
1	1	15	16
2	4	21	25
3	9	27	36
4	16	33	49

(a)
Patterns for the number of shaded triangles
Pattern 1: 1 x 6 + 9 = 15
Pattern 2: 2 x 6 + 9 = 21
Pattern 3: 3 x 6 + 9 = 27
Pattern 4: 4 x 6 + 9 = 33

Number of shaded triangles = Pattern number x 6 + 9
Pattern number = (Number of shaded triangles - 9) ÷ 6

Pattern number for 87 shaded triangles
= (87 - 9) ÷ 6
= 78 ÷ 6
= 13

(b)
Pattern for the number of unshaded triangles
Pattern 1: 1² = 1
Pattern 2: 2² = 4
Pattern 3: 3² = 9
Pattern 4: 4² = 16

Number of shaded triangles in Pattern 41
= 41²
= 1681

(c)
Pattern for the total number of triangles
Pattern 1: (1 + 3)² = 16
Pattern 2: (2 + 3)² = 25
Pattern 3: (3 + 3)² = 36
Pattern 4: (4 + 3)² = 49

Total number of triangles = (Pattern number + 3)²

Total number of triangles for Pattern 65
= (65 + 3)²
= 68²
= 4624

Answer(s): (a) 13; (b) 1681; (c) 4624