Some patterns of shaded and unshaded small squares is shown. The unshaded squares are those which lie on the diagonals of the figure. By considering the number patterns,

1. find the total number of squares in Figure 31,
2. find the total number of shaded squares in Figure 59.
3. what will be the figure number that will have 25 squares?

Pattern for number of squares
Figure 1: (1 x 2 - 1)² = 1
Figure 2: (2 x 2 - 1)² = 9
Figure 3: (3 x 2 - 1)² = 25
Figure 4: (4 x 2 - 1)² = 49

Number of squares = (Figure number x 2 - 1)²

Number of squares for Figure 31
= (31 x 2 - 1)²
= 61 x 61
= 3721

(b)
Pattern for the number of shaded squares
Figure 1: ((1 - 1) x 2)² = 0
Figure 2: ((2 - 1) x 2)² = 4
Figure 3: ((3 - 1) x 2)² = 16
Figure 4: ((4 - 1) x 2)² = 36

Number of shaded squares = ((Figure number - 1) x 2)²

Number of shaded squares for Figure 59
= ((59 - 1) x 2)²
= 116 x 116
= 13456

(c)
Number of squares = (Figure number x 2 - 1)²
Figure number = {(√ Number of squares) + 1} ÷ 2

Figure number with 25 squares
= {(√25 + 1) ÷ 2
= (5 + 1) ÷ 2
= 6 ÷ 2
= 3

Answer(s): (a) 3721; (b) 13456; (c) 3