PSLE
Jane uses stars and circles to form figures that follow a pattern as shown.

1. The table shows the number of stars and circles for the first four figures. Complete the table for Figure 8. List the answers in the following order using commas. (Eg i, ii, iii)
2. A figure in the pattern has a total of 64 stars and circles. What is the figure number?
3. Another figure in the pattern has 31 more stars than circles. What is the total number of stars and circles in the figure?

(a)
Pattern of the number of stars:
Figure 1: 2 = 2
Figure 2: 2 + 3 = 5
Figure 3: 2 + 3 + 4 = 9
Figure 4: 2 + 3 + 4 + 5 = 14

To find the total number of stars in Figure 8, the last digit in the string of addition is 1 more than 8 which is 9.

Figure 8: 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 44

Total number of stars for Figure 8 = 44 (i)

Pattern of the number of circles:
Figure 1: 2 = 2
Figure 2: 2 + 2 = 4
Figure 3: 2 + 2 + 3 = 7
Figure 4: 2 + 2 + 3 + 4 = 11

To find the total number of circles in Figure 8, the last digit in the string of addition is 8.

Figure 8: 2 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 37

Total number of circles for Figure 8 = 37 (ii)

Total number of stars and circles for Figure 8
= 44 + 37
= 81 (iii)
 
(b)
Pattern of the total number of stars and circles:
Figure 1: 2² = 2 x 2 = 4
Figure 2: 3² = 3 x 3 = 9
Figure 3: 4² = 4 x 4 = 16
Figure 4: 5² = 5 x 5 = 25
Figure 5: 6² = 6 x 6 = 36

Formula:
Total number of stars and circles = (Figure number + 1)²

64 = 8 x 8
√64 = 8

Figure number that has a total number of 64 stars and circles
= 8 - 1
= 7

(c)
Pattern of the number of more stars than circles:
Figure 1: 2 - 2 = 0
Figure 2: 5 - 4 = 1
Figure 3: 9 - 7 = 2
Figure 4: 14 - 11 = 3

Figure number = Number of stars more than circles + 1

Figure number that has 31 more stars than circles
= 31 + 1
= 32

Total number of stars and circle for Figure 32
= (32 + 1)²
= 33 x 33
= 1089

Answer(s): (a) 44, 37, 81; (b) 7; (c) 1089