# Worked Examples

Observe the numbers in the opposite sectors of the circle.

They are of the form n, n^{2} + 5.

30 = 5^{2} + 5; 41 = 6^{2} + 5. So, the missing number is 7^{2} + 5 = 54.

Answer: 54

On analysing the pattern of numbers in the first figure, we notice that the number inside the triangle is related to the numbers at the vertices by the following formula:

Formula:

By applying the same formula, we can find the number inside the second triangle.

Answer: 10

We notice that the outer square is divided into four equal smaller squares and each such smaller square contains three numbers. These three numbers are related as given below:

15 Ã— 10 Ã· 2 = 75

14 Ã— 17 Ã· 2 = 119

16 Ã— 12 Ã· 2 = 96

By applying the same formula, we can find the missing number.

18 Ã— 11 Ã· 2 = 99

Answer: 99

Each of the four smaller squares contains three numbers. The innermost number is half the square of the sum of the numbers that are outside.

5 + 11 = 16; 16^{2} = 256 256 Ã· 2 = 128

9 + 5 = 14; 14^{2} = 196 196 Ã· 2 = 98

11 + 7 = 18; 18^{2} = 324 324 Ã· 2 = 162

8 + 4 = 12; 12^{2} = 144 144 Ã· 2 = 72

Answer: 72

The number in the inner small square is twice the sum of the square roots of the outer numbers.

18 Ã— 2 = 36

11 Ã— 2 = 22

14 Ã— 2 = 28

17 Ã— 2 = 34

Answer: 34

Consider the numbers at the ends of each straight line segments. They are related to each other as under:

4 Ã— 3 = 12

6 Ã— 4 = 24

7 Ã— 5 = 35

9 Ã— 6 = 54

Answer: 54

Add the outer numbers in the bigger circle and multiply the sum by the number at the top to obtain the number inside the small circle.

(7 + 5 + 4) Ã— 7 = 16 Ã— 7 = 112

Similarly, (8 + 3 + 6) Ã— 8 = 17 Ã— 8 = 136

Answer: 136

The number inside the circle is the square root of the sum of all the numbers that are outside.

14 + 17 + 9 + 6 + 11 + 7 = 64; = 8

12 + 9 + 14 + 15 + 13 + 18 = 81; = 9

Answer: 9

Observe the numbers in the opposite segments of the hexagon. They follow the pattern n, n^{3}.

3, 3^{3} = 27 5, 5^{3} = 125 6, 6^{3} = 216

Similarly, 4, 4^{3} = 64 7, 7^{3} = 343 8, 8^{3} = 512

Answer: 512

The numbers in the opposite segments of the circle follow the pattern n, n^{3} + 1.

3, 3^{3} + 1 = 28; 5, 5^{3} + 1 = 126

2, 2^{3} + 1 = 9

Similarly, 4, 4^{3} + 1 = 65; 8, 8^{3} + 1 = 513

7, 7^{3} + 1 = 344

Answer: 344