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 31,
- find the total number of shaded squares in Figure 34.
- what will be the figure number that will have 100 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 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 34
= ((34 - 1) x 2)
2 = 66 x 66
= 4356
(c)
Number of squares = (Figure number x 2 - 1)
2 Figure number = {(√ Number of squares) + 1} ÷ 2
Figure number with 100 squares
= {(√100 + 1) ÷ 2
= (10 + 1) ÷ 2
= 11 ÷ 2
= 5.5
Answer(s): (a) 3721; (b) 4356; (c) 5.5