Riordan formed some figures using squares and circles as shown.

1. Find the number of squares in Figure 10.
2. Find the total number of circles and squares in Figure 26.
3. A figure in the pattern has a total of 4291 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 10
= 1 + 10
= 11

(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 26 x 27)
= 1 + 26 x 27
= 1 + 702
= 703

(c)
1 + 2[¹₂ x Figure Number x (Figure Number + 1)] = 4291
2[¹₂ x Figure Number x (Figure Number + 1)] = 4291 - 1
Figure Number x (Figure Number + 1) = 4290
65 x 66 = 4290

Figure number that has a total number of circles and squares = 65

Answer(s): (a) 11; (b) 703; (c) 65