A pattern is made by Billy putting shaded squares of unit length around a white square of similar unit length. He then continues to make patterns as shown in Figure 2 and Figure 3.
- Which figure uses 121 white squares?
- Find the number of shaded squares in Figure 21.
- Find the total number of squares in Figure 51.
Pattern |
|
|
|
|
|
|
Figure |
1 |
2 |
3 |
(a) ? |
21 |
51 |
Number of white squares |
1 |
4 |
9 |
121 |
|
|
Number of shaded squares |
8 |
12 |
16 |
|
(b) ? |
|
Total number of squares |
9 |
16 |
25 |
|
|
(c) ? |
(a)
Pattern for the number of white squares
Figure 1: 1
2 = 1 x 1 = 1
Figure 2: 2
2 = 2 x 2 = 4
Figure 3: 3
2 = 3 x 3 = 9
Figure 4: 4
2 = 4 x 4 = 16
Number of white squares
= Figure number x Figure number
= Figure number
2Figure number that uses 121 white squares
= √121
= 11
(b)
Patterns for the number of shaded squares
Figure 1: 1 x 4 + 4 = 8
Figure 2: 2 x 4 + 4 = 12
Figure 3: 3 x 4 + 4 = 16
Formula:
Number of shaded squares = Figure Number x 4 + 4
Number of shaded squares in Figure 30
= 21 x 4 + 4
= 88
(c)
Pattern for the total number of squares
Figure 1: (1 + 2)
2 = 9
Figure 2: (2 + 2)
2 = 16
Figure 3: (3 + 2)
2 = 25
Total number of squares = (Figure number + 2)
2 Total number of squares for Figure 51
= (51 + 2)
2= 53 x 53
= 2809
Answer(s): (a) 11; (b) 88; (c) 2809