Reggie formed some figures using squares and circles as shown.

1. Find the number of squares in Figure 12.
2. Find the total number of circles and squares in Figure 27.
3. A figure in the pattern has a total of 7311 circles and squares. What is the Figure Number?

(a)
Number of squares
Figure 1: 2 = 1 + 1
Figure 2: 3 = 1 + 2
Figure 3: 4 = 1 + 3
Figure 4: 5 = 1 + 4

Formula:
Number of squares = 1 + Figure Number

Number of squares in Figure 12
= 1 + 12
= 13

(b) 
Total number of circles and squares
Figure 1: 3   = 1 + 2(1)
Figure 2: 7   = 1 + 2(1 + 2)
Figure 3: 13 = 1 + 2(1 + 2 + 3)
Figure 4: 21 = 1 + 2(1 + 2 + 3 + 4)

Formula:
Number of circles and squares= 1 + 2[¹₂ x Figure Number x (Figure Number + 1)]

Total number of circles and squares
= 1 + 2[¹₂ x Figure Number x (Figure Number + 1)]
= 1 + 2(¹₂ x 27 x 28)
= 1 + 27 x 28
= 1 + 756
= 757

(c)
1 + 2[¹₂ x Figure Number x (Figure Number + 1)] = 7311
2[¹₂ x Figure Number x (Figure Number + 1)] = 7311 - 1
Figure Number x (Figure Number + 1) = 7310
85 x 86 = 7310

Figure number that has a total number of circles and squares = 85

Answer(s): (a) 13; (b) 757; (c) 85