The first three figures of a sequence are shown. The table records the number of shaded and unshaded rhombuses in each figure.

1. Find the number of unshaded rhombuses in Figure 12?
2. Find the number of shaded rhombuses in Figure 55?
3. Which figure number has 672 shaded rhombuses?

Figure number	Number of unshaded rhombuses	Number of shaded rhombuses
1	1	8
2	4	12
3	9	16

(a)
Pattern for the number of unshaded rhombuses
Figure 1: 1² = 1
Figure 2: 2² = 4
Figure 3: 3² = 9

Number of unshaded rhombuses = Figure number²

Number of unshaded rhombuses in Figure 12
= 12²
= 144

(b)
Pattern for the number of shaded rhombuses
Figure 1: 1 x 4 + 4 = 8
Figure 2: 2 x 4 + 4 = 12
Figure 3: 3 x 4 + 4 = 16

Number of shaded rhombuses = Figure number x 4 + 4

Number of shaded rhombuses in Figure 55
= 55 x 4 + 4
= 220 + 4
= 224

(c)
Number of shaded rhombuses = Figure number x 4 + 4
Figure number = (Number of shaded rhombuses - 4) ÷ 4

Figure number with 672 shaded rhombuses
= (672 - 4) ÷ 4
= 167

Answer(s): (a) 144; (b) 224; (c) 167