# Workout the Mystery Number using Squaring

In this worksheet, students will identify a rule which is used to make a number and apply this rule to find a missing number.

Key stage:  KS 2

Curriculum topic:   Verbal Reasoning

Curriculum subtopic:   Number Logic

Difficulty level:

### QUESTION 1 of 10

Hi number detective! We’ve got a new challenge that we need your help with. Are you ready?

We need to work out how the numbers on the outside of the brackets create the numbers on the inside of the brackets and then apply the same rule to the third set of brackets.

10 (23) 3        5 (16) 6         2 ( __) 5

Be warned:

One of the numbers could be used more than once.

A third invisible number could be used.

One or both of the numbers could be squared (multiplied by themselves).

First, we need to find a way to make 23 using 10 and 3. Remember that we will probably have to consider one of the warnings above.

If we add 10 and 3 together we get 13. Then, if we add another lot of 10, we get to 23.

So the rule could be: add the two outside numbers and then add the first number again.

Let’s try this out for the second set of numbers: 5 + 5 + 6 = 16.

We’ve found the rule! Jackpot!

Let’s use this rule to work out the missing number in the final set: 2 + 2 + 5 = 9. Therefore, the missing number is 9.

Let’s try another:

5 (28) 3          3 (16) 7          4 ( __ ) 8

First, we need to find a way to make 28 using 5 and 3. Remember that we can use one of the warnings listed above.

Let’s try 5 + 3 = 8 and add the invisible number of 20 to get to 28.

So our rule could be: add the two numbers together and then add the invisible number 20.

Let’s see if this works for the second set of numbers: 3 + 7 = 10 but we need to use a different invisible number of 8 to get to our total.

Bad news, our rule didn’t work so we need to find another.

Let’s try (this means 5 x 5) which gives us 25 and then we need to add 3 to get to 28.

So our rule could be: square the first number and then add the second.

Let’s try this for the second set: is 9 then add 7 gives us 16.

Jackpot! We’ve found the correct rule.

Let’s use this rule to work out the missing number in the final set of numbers: 4² = 16 and then add 8 gives us 24. So the correct answer is 24.

 Pssstt! Here's a handy hint to help you reach superstar status: Look out for the secret number and don't worry if the first rule you find doesn't work, just stay calm and try another one!

It’s now your turn to be a number detective and hunt down those missing numbers!

Welcome to another exciting activity! I've got some number puzzles that I need your help with, detective. I hope you have your problem-solving hat on!

Look at the groups of numbers below:

6 (43) 7       9 (37) 4

How do the outside numbers in each set create the numbers inside the brackets?

Left number x right number

Left number - right number

(Left number x right number ) + 1

Left number + right number + 4

Ready for another one, number detective?

Look at the numbers below:

Which rule from below creates the number inside the brackets by using the two outside numbers? Make sure that the rule you select works for both number sets.

4 (49) 3        7 (144) 5

Left number x right number x 2

(Left number x right number) + 7

(Left number + right number)²​

Left number - right number

I think you're ready to work out some mystery numbers!

Look at the sets of numbers below:

6 (21) 7        4 (10) 5

The rule here is '(left number x right number) ÷ 2.'

This means that we multiply the left number by the right number and then divide that answer by 2 to get the middle number.

Use this rule to work out the mystery number in the middle of the brackets below.

Left number x right number x 2

(Left number x right number) + 7

(Left number + right number)²​

Left number - right number

Look at the number sets below:

6 (31) 5        10 (93) 7

The rule here is 'left number squared - right number'.

Use this rule to work out the mystery number in the brackets below.

Left number x right number x 2

(Left number x right number) + 7

(Left number + right number)²​

Left number - right number

You're halfway there already, number detective!

Look at the number sets below:

7 (27) 10        9 (37) 14

The rule here is 'left number + right number + right number'.

Use this rule to work out the mystery number in the brackets below.

Left number x right number x 2

(Left number x right number) + 7

(Left number + right number)²​

Left number - right number

It's time to make this a bit more challenging!

This time you will need to work out the rule for the first two sets of numbers and then use this rule to work out the mystery number in the final set.

3 (28) 6        4 (46) 9        7 (?) 3

Look at the sets of numbers below.

Work out the rule that creates the numbers inside the brackets by using the two outside numbers.

Then use this rule to work out the mystery number in the third set.

6 (7) 10        20 (25) 14        9 (?) 10

Look at the numbers below:

12 (152) 8        9 (89) 8        7 (?) 10

Work out the rule which makes the middle number in the first two sets and use this rule to work out the mystery number in the final set.

70

77

59

149

Look at the numbers below:

108 (12) 9        45 (15) 3        84 (?) 4

Work out the rule which creates the middle number in the first two sets and use this rule to work out the mystery number in the final set.

80

336

76

21

It's time for the final number puzzle of this activity!

Look at the number sets below:

7 (13) 3       9 (33) 12        6 (?) 13

Work out the rule for the first two number sets and use this rule to calculate the mystery number in the final set.

32

18

78

7

• Question 1

Welcome to another exciting activity! I've got some number puzzles that I need your help with, detective. I hope you have your problem-solving hat on!

Look at the groups of numbers below:

6 (43) 7       9 (37) 4

How do the outside numbers in each set create the numbers inside the brackets?

(Left number x right number ) + 1
EDDIE SAYS
Hey number detective! If you struggle to see the pattern here at first, you can try out each of the options until you find one that works for both sets. The rule here was '(left number x right number) + 1'. Let's look at how this rule works for the first set: 6 (43) 7 6 (left number) x 7 (right number) = 42. 42 + 1 = 43 (middle number). This also works for the second set: 9 (37) 4 9 x 4 = 36 36 + 1 = 37 which is our middle number. Top tip - make sure that the rule works for all sets of numbers.
• Question 2

Ready for another one, number detective?

Look at the numbers below:

Which rule from below creates the number inside the brackets by using the two outside numbers? Make sure that the rule you select works for both number sets.

4 (49) 3        7 (144) 5

(Left number + right number)²​
EDDIE SAYS
Did you notice that the numbers inside the brackets here are squared numbers? This is a clue that our rule will have something to do with squaring. The rule is '(left number + right number)²​'. This means that we add the left and right numbers together and then square the answer. A little ²​ means squared. Let's look at how this works for the first set of numbers: 4 (49) 3 4 + 3 = 7 and 7²​ = 49. Can you see how it works for the second set also?
• Question 3

I think you're ready to work out some mystery numbers!

Look at the sets of numbers below:

6 (21) 7        4 (10) 5

The rule here is '(left number x right number) ÷ 2.'

This means that we multiply the left number by the right number and then divide that answer by 2 to get the middle number.

Use this rule to work out the mystery number in the middle of the brackets below.

EDDIE SAYS
Did you follow the rule for this next set of numbers? We need to multiply 6 (left number) by 8 (right number) to get 48. We then need to divide this number by 2: 48 ÷ 2 = 24. Do you feel like you're at school with all these rules?
• Question 4

Look at the number sets below:

6 (31) 5        10 (93) 7

The rule here is 'left number squared - right number'.

Use this rule to work out the mystery number in the brackets below.

EDDIE SAYS
Did you follow the rule to get 14 here? We need to square the number on the left of the brackets: 4 (?) 2 4² = 16 and then we need to subtract the right number. 16 - 2 = 14. This means 14 is our mystery number!
• Question 5

You're halfway there already, number detective!

Look at the number sets below:

7 (27) 10        9 (37) 14

The rule here is 'left number + right number + right number'.

Use this rule to work out the mystery number in the brackets below.

EDDIE SAYS
Let's follow the rule carefully here: 8 (?) 15 8 (left number) + 15 (right number) + 15 (right number) = 38. OK detective, I think you're ready for your next challenge...
• Question 6

It's time to make this a bit more challenging!

This time you will need to work out the rule for the first two sets of numbers and then use this rule to work out the mystery number in the final set.

3 (28) 6        4 (46) 9        7 (?) 3

31
thirty one
thirty-one
EDDIE SAYS
Did you find a connection between the numbers on the outside of the brackets and the numbers on the inside of the brackets? The number in brackets is quite a lot bigger than the outside numbers. This is a clue that we are multiplying or squaring the numbers. Let's look at multiplying in the first set: 3 (28) 6 3 x 6 = 18 How can we get from 18 to 28? We add a sneaky invisible 10. So the rule could be 'left number x right number + 10'. Does it work with the second set? 4 (46) 9 4 x 9 = 36 and 36 + 10 = 46. Jackpot it works! Now we can use this rule to work out the mystery number. 7 (?) 3 7 x 3 = 21 and 21 + 10 = 31. This makes 31 our mystery number. Keep looking out for those multiplication clues, detective.
• Question 7

Look at the sets of numbers below.

Work out the rule that creates the numbers inside the brackets by using the two outside numbers.

Then use this rule to work out the mystery number in the third set.

6 (7) 10        20 (25) 14        9 (?) 10

10
ten
EDDIE SAYS
Don't worry if the first rule you make doesn't work for both the first two sets of numbers as this happens to a lot of people. The trick is to not give up and keep trying out different rules until one works for both sets. Let's look at the first set: 6 (7) 10 6 + 10 = 16 and if we subtract 9, we get the middle number (7). The rule could be 'left number + right number - 9'. Does it work for the second set? 20 (25) 14 20 (left number) + 14 (right number) = 34 and if we subtract 9 from 34, we get 25: 25 is the middle number. Yes, it works! Now we can use this rule to work out the mystery number: 9 (?) 10 9 + 10 = 19 and subtract 9 = 10 This makes 10 our mystery number!
• Question 8

Look at the numbers below:

12 (152) 8        9 (89) 8        7 (?) 10

Work out the rule which makes the middle number in the first two sets and use this rule to work out the mystery number in the final set.

59
EDDIE SAYS
We had to square in our rule here! The rule is 'left number squared + right number'. Let's look at how this works for the first set of numbers: 12 (152) 8 As the number inside the brackets is a lot bigger than the outside numbers, it's a clue that we are multiplying or squaring. Remember to look out for these clues, detective. 12² = 144 and 144 + 8 (right number) = 152. Does it work for the second set too? 9 (89) 8 9² (81) + 8 = 89. It does work! Now that we've checked that the rule works for both sets, we can work out the mystery number: 7 (?) 10 7² (49) + 10 = 59 This makes 59 our mystery number.
• Question 9

Look at the numbers below:

108 (12) 9        45 (15) 3        84 (?) 4

Work out the rule which creates the middle number in the first two sets and use this rule to work out the mystery number in the final set.

21
EDDIE SAYS
The numbers inside the brackets are a lot smaller than the numbers on the outside. This is a clue that we're dividing. If we were subtracting, the numbers wouldn't have shrunk that much! Let's look at the first set: 108 (12) 9 Is there a way to instantly get an answer of 12 by using the numbers 108 and 9? Yes! We can divide: 108 ÷ 9 = 12 The rule here could be 'left number ÷ right number'. Does it work for the next set? 45 (15) 3 You might have to do some quick working out on a piece of paper here to see that 45 ÷ 3 = 15. The rule works! Now we can work out the mystery number: 84 (?) 4 84 ÷ 4 = 21 21 is our mystery number. You can do short division (bus stop method) on a piece of paper to get the answer as quickly as possible.
• Question 10

It's time for the final number puzzle of this activity!

Look at the number sets below:

7 (13) 3       9 (33) 12        6 (?) 13

Work out the rule for the first two number sets and use this rule to calculate the mystery number in the final set.

32
EDDIE SAYS
In these sets of numbers, we used one of the numbers twice. The rule was 'left number + right number + right number again'. Can you see how this works for the first set of numbers? 7 (13) 3 7 + 3 + 3 = 13. It also works for the second set too, so now we can work out the mystery number: 6 (?) 13 6 (left number) + 13 (right number) + 13 (right number again) = 32 This makes 32 our mystery number! Well done for all of your awesome number skills in this activity, detective! It's been great working with you. You have developed your ability to identify mystery numbers using more advanced operations such as squaring!
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