__Objective:__

At the end of the lesson, students should be able to solve number puzzles.

Many puzzles using addition and subtraction have developed over the years. One well known puzzle is the **magic square.**

Magic squares are a kind of puzzle in which numbers are put in small squares and their sum across the squares should match up to their sum along the square and along the diagonals.

__Example 1__

**Complete this magic square**

__Solution __

**The magic number here is 15**

The example above shows that the total along any straight line is equal to 15. The total is called the **magic number.**

__Example 2__

Find the value of A and B if

__Solution __

Since each letter represents the same digit throughout the coded puzzle we could try A = 1 therefore 11 1 = 11 which means the answer would also be AA.

If we try A = 2 we get 22 2 = 44

So A = 2 and B = 4.

__Example 3__

Which number will replace the question mark?

__Solution __

In each row there are ‘**A**‘, ‘**B**‘ and ‘**C**‘

In the second row ‘**A**‘ and ‘**C**‘ are already there

Hence in place of?, there will be ‘B’.

From first row: **4A** x **6C** = **24B**

From third row: **9B **x **4C** = **36A**

From second row: **5A** x ? = **45C**

? = (45C/5A)

? = **9B**

Hence the number **9B** will replace the question mark.

__Example 4__

If ‘AND’ is written as ‘EQF’ and ‘THE’ as ‘XKG’ then how will ‘COM’ be written?

Solution

if AND is written as,

Similarly, ‘COM’ be written as;

__Example 5__

Which number will replace the question mark?

__Solution __

From fig. a: 6 + 4 + 8 = 18

18 + 2 = 20

From fig. b: 7 + 9 + 8 = 24

24 + 2 = 26

From fig. c: 6 + 5 + 12 = 23

23 + 2 = **25**

Hence the number **25** will replace the question mark.

__Example 6__

Which number will replace the question mark?

__Solution__

From fig. a: (3)^{2} + (2)^{2} = 13

From fig. b: (4)^{2} + (8)^{2} = 80

From fig. c:? = (1)^{2} + (5)^{2}

? = 1 + 25

? = **26**

Hence the number **26** will replace the question mark.

__Example 7__

Which number will replace the question mark?

__Solution __

From fig. a: 9^{2} + 8^{2} + 7^{2} + 6^{2} = 81 + 64 + 49 + 36 = 230

From fig. b: 6^{2} + 7^{2} + 3^{2} + 4^{2} = 36 + 49 + 9 + 16 = 110

From fig. c: 9^{2} + 6^{2} + 5^{2} + 4^{2} = 81 + 36 + 25 + 16 = **158**

Hence the number **158** will replace the question mark.

__Example 8__

Which number will replace the question mark?

__Solution __

(4 + 3)^{2} = (7)^{2} = 49

(8 + 5)^{2} = (13)^{2} = 169

(11 + 12)^{2} = (23)^{2} = 529

(10 + 9)^{2} = (19)^{2} = **361**

Hence the number **361** will replace the question mark.

__Example 9__

Which number will replace the question mark?

__Solution __

From fig. a: (8 x 5) – (4 x 3) = 40 – 12 = 28

From fig. b: (12 x 7) – (8 x 9) = 84 – 72 = 12

From fig. c: (5 x 3) – (6 x ?) = 21

15 – 6? = 21

6? = -6

? = **-1**

Hence the number **-1** will replace the question mark.

__Example 10__

__Solution __

**The answer is Q**

Adding the three numbers in each square together gives the numerical value of the letter at the centre of each square

__Example 11__

__Solution__

**The answer is 22 **

Add together values in corresponding positions of the top two crosses, and put the results in the lower left cross. Calculate the difference between values in corresponding positions of the top two crosses, and put the results in the lower right cross. Finally, add together the values in corresponding positions of the lower two crosses to give the values in the central cross.

__Strategies For Solving Puzzle Problems__

- Read the question at least twice, it is always a good start to make sure you understand the problem.
- Then read the question again.
- Look for patterns – each puzzle is usually different so you have to look for the patterns that apply to each puzzle problem.
- Guess and check.
- Work backwards if possible.
- Solve the puzzle.

__Unit 1 Lesson 5: Exercise 1__

CHECK ANSWERS AT LESSON MATERIAL ( AT THE TOP OF THE LESSON )

5) The answer is 0.

Explanation: Looking at the line of numbers from top;

9 x 8 = 72

72 x 8 = 576

576 x 8 = 4608

6) Answer is 4.

Explanation: Looking across the three circles, the number in the middle is the product of the two numbers in the same segment in the other two circles. i.e;

3 x 2 = 6, 7 x 3 = 21 and 4 x 4 = 16.

8) Answer is 13.

Explanation: Numbers follows the alternate of prime numbers

3 + 7 + 9 + 12 + 10 = 41

4 + 6 + 10 + 9 = 31

7 + 9 + 7 = 23

5 + 12 = 17

So the next prime number is 13 then 11, etc.

9) Answer is 49.

Explanation: Add the two oval and divide the result by two to get the bigger oval, i.e. (47 + 63) ÷ 2 = 55, (85 + 99) ÷ 2 = 92, (73 + 25) ÷ 2 = ?, ? = 49

10) The answer is 1.

Explanation: Looking at the line of number from the top two boxes add up to give the next box below it. i.e. 6 + 7 = 13 and 7 + 2 = 9

Next row; 13 + 9 = 22

Fourth row; 17 + 5 = 22

Last row; 13 + 4 = 17

4 + ? = 5

Therefore, 5 – 4 = ?

? = 1.

Lesson Content

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