Elyse added a certain number of white squares round a grey square and then a certain number of grey squares round the white squares and she continued adding more grey and white squares round the diagram in each pattern as shown.
- How many grey squares did she need if the most outer layer had 15 squares on each side?
- How many squares are there in the Figure 16?
(a)
Formula:
Number of squares on each side = Figure number x 2 - 1
Figure number x 2 - 1 = 15
Figure number x 2 = 15 + 1 = 16
Figure number = 16 ÷ 2 = 8
Pattern for the total number of squares
Figure 1: 1 x 1 = 1 square
Figure 2: 3 x 3 = 9 squares
Figure 3: 5 x 5 = 25 squares
Formula:
Total number of squares = [(Figure number x 2) - 1]
2
Total number of squares at Figure 8
= ((8 x 2) - 1)
2 = 15 x 15
= 225
Figure 5 has 32 white squares (same as Figure 4)
Figure 6 has 40 white squares on its outermost layer.
Number of white squares at Figure 6
= 32 + 40
= 72
Number of white squares at Figure 7 = 72 (same as Figure 6)
Figure 8 has 56 white squares on its outermost layer.
Number of white squares at Figure 8
= 72 + 56
= 128
Number of grey squares at Figure 8
= Total number of squares at Figure 8 - Number of white squares at Figure 8
= 225 - 128 = 97
(b)
Total number of squares at Figure 16
= [(16 x 2) - 1]
2 = 31 x 31
= 961
Answer(s): (a) 97; (b) 961