Riordan formed some figures using squares and circles as shown.

1. Find the number of squares in Figure 9.
2. Find the total number of circles and squares in Figure 24.
3. A figure in the pattern has a total of 651 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 9
= 1 + 9
= 10

(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 24 x 25)
= 1 + 24 x 25
= 1 + 600
= 601

(c)
1 + 2[¹₂ x Figure Number x (Figure Number + 1)] = 651
2[¹₂ x Figure Number x (Figure Number + 1)] = 651 - 1
Figure Number x (Figure Number + 1) = 650
25 x 26 = 650

Figure number that has a total number of circles and squares = 25

Answer(s): (a) 10; (b) 601; (c) 25