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,
- find the total number of squares in Figure 37,
- find the total number of shaded squares in Figure 56.
- what will be the figure number that will have 64 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)
2Number of squares for Figure 37
= (37 x 2 - 1)
2 = 73 x 73
= 5329
(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 56
= ((56 - 1) x 2)
2 = 110 x 110
= 12100
(c)
Number of squares = (Figure number x 2 - 1)
2 Figure number = {(√ Number of squares) + 1} ÷ 2
Figure number with 64 squares
= {(√64 + 1) ÷ 2
= (8 + 1) ÷ 2
= 9 ÷ 2
= 4.5
Answer(s): (a) 5329; (b) 12100; (c) 4.5