About Number Puzzles
A number puzzle gives several rows in which the third number is worked out from the first two in the same way, and asks you to apply that rule to a new row. The rule has to fit every complete row, and a well-set puzzle has exactly one rule that does, so the work is to find it and then check it against all the rows rather than the first one.
What you need to understand
- The rule must fit every complete row, not just the first. Testing a single row is how a wrong rule gets chosen.
- Compare the third number with the first two: much larger suggests multiplication, only slightly different suggests addition or subtraction.
- Halving is easy to overlook because it keeps the third number close to the inputs.
- A rule with one added on is easily confused with plain multiplication when the first two numbers are small.
- Subtraction depends on the order of the two numbers, so test it both ways round.
- Once a rule fits, apply it to the question row and check the result is the same size as the other third numbers.
How to work through these questions
- Take the first row and guess a simple rule that turns the first two numbers into the third.
- Test that guess against every other complete row before doing anything else.
- Discard it at the first row it fails and guess again, this time from a row you have not tried.
- When a rule survives every complete row, apply it to the numbers in the question row.
- Check the answer is the same size and sign as the other third numbers.
- If two rules both fit, the puzzle is ambiguous; prefer the simpler and say why.
Mistakes that cost marks
- Choosing a rule from the first row and never testing it against the others.
- Overlooking halving, which keeps numbers small and is easy to miss.
- Trying only multiplication and addition, and never subtraction.
- Using a rule with one added on when plain multiplication also fits every row.
- Applying the rule to the numbers in the wrong order when it involves subtraction.
- Ignoring one row that does not fit, when a single bad row means the rule is wrong.
Worked example
In each row the third number is obtained from the first two in the same way. 3 and 4 give 11; 11 and 12 give 131; 4 and 12 give 47. What do 8 and 10 give?
- Each third number is much larger than either of the first two, so the rule multiplies.
- 3 x 4 = 12 while the third number is 11, so try multiplying and then subtracting one.
- Test it on the second row: 11 x 12 = 132, and 132 - 1 = 131, which matches.
- Test it on the third row: 4 x 12 = 48, and 48 - 1 = 47, which matches.
- So 8 x 10 = 80, and 80 - 1 = 79.
Answer: 79
Practice questions with answers
A few Number Puzzles questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
In each row the third number is obtained from the first two in the same way.
5 and 11 give 14
2 and 5 give 5
10 and 4 give 12
8 and 4 give ?
Answer: Option D — with explanation
Only one rule fits all 3 complete rows, and it is to add the first two and then subtract two.
So 8 and 4 give 10.
Common mistakes
- used the rule to double the sum of the first two: 24
- used the rule to multiply the first two and then subtract two: 30
- used the rule to add the first two and then subtract one: 11
Question 2
In each row the third number is worked out from the first two by the same rule.
3 and 4 give 13
11 and 12 give 133
4 and 12 give 28
8 and 10 give 74
4 and 9 give ?
Answer: Option B — with explanation
Only one rule fits all 4 complete rows, and it is to square the first and add the second.
So 4 and 9 give 25.
Common mistakes
- used the rule to multiply the first two and then add one: 37
- used the rule to add the first two and then add two: 15
- used the rule to multiply the first two and then subtract both numbers: 23
Question 3
Each row follows one rule that turns the first two numbers into the third.
6 and 6 give 13
4 and 5 give 10
12 and 9 give 22
5 and 3 give 9
7 and 5 give 13
11 and 9 give ?
Answer: Option C — with explanation
Only one rule fits all 5 complete rows, and it is to add the first two and then add one.
So 11 and 9 give 21.
Common mistakes
- used the rule to add the first two and then subtract two: 18
- used the rule to square the first and add the second: 130
- used the rule to multiply the first two: 99
Question 4
Study the rows: in each one the third number comes from the first two in the same way.
5 and 9 give 34
6 and 3 give 39
9 and 8 give 89
4 and 2 give ?
Answer: Option B — with explanation
Only one rule fits all 3 complete rows, and it is to square the first and add the second.
So 4 and 2 give 18.
Common mistakes
- used the rule to double the sum of the first two: 12
- used the rule to add the first two and then subtract two: 4
- used the rule to multiply the first two and then subtract one: 7
Question 5
In each row the third number is obtained from the first two in the same way.
8 and 6 give 34
3 and 7 give 11
8 and 2 give 6
8 and 5 give ?
Answer: Option A — with explanation
Only one rule fits all 3 complete rows, and it is to multiply the first two and then subtract both numbers.
So 8 and 5 give 27.
Common mistakes
- used the rule to subtract the second from the first: 3
- used the rule to add the first two and then subtract one: 12
- used the rule to double the first and add the second: 21
Frequently asked questions
How do I know which rule to try first?
Compare the third number with the first two. A much larger third number suggests multiplication, while one close to the inputs suggests addition or subtraction.
What if a rule fits the first row but not the second?
Then it is the wrong rule. A rule has to fit every complete row, and one failure is enough to discard it.
What if more than one rule fits all the rows?
Then the puzzle is not well set, because there is more than one defensible answer. A properly made puzzle has exactly one simple rule that fits every row.
Is halving allowed as a rule?
Yes, and it is easy to miss because it keeps the third number near the inputs. Look for it whenever the third number is about half the product.
Does the order of the first two numbers matter?
It does for subtraction, so test such a rule both ways round before accepting it.
