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

1. Which pattern has 63 shaded triangles?
2. Find the number of unshaded triangles in Pattern 33.
3. Find the total number of triangles in Pattern 33.

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 63 shaded triangles
= (63 - 9) ÷ 6
= 54 ÷ 6
= 9

(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 33
= 33²
= 1089

(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 33
= (33 + 3)²
= 36²
= 1296

Answer(s): (a) 9; (b) 1089; (c) 1296