Missing Number

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How missing-number puzzles work

⏱ 5 min read🧩 6 question types🎯 12 practice Q

The idea in one minute

A figure hides one number with a '?'. The other parts of the figure follow one hidden rule. Find the rule from the complete rows, check it on every row, then use it once on the row with the '?'. Most rules are simple: add, multiply, square, or one small change on top.

What the puzzle looks like

A figure hides one number. It may be rows of numbers, two square grids, a circle or a triangle. Every complete part follows the same rule. Find it. Then use it once on the part with the "?".

Small example: 3 → 9, 4 → 16, 5 → ?. Each number is squared. So ? = 25.

Rule: The rule is the same in every complete row. One row can fit a wrong rule by luck. Two rows almost never do.

The rule ladder

Test simple rules first. Stop at the first rule that fits all the rows.

Step	Rule to test	Sample with 4 and 7
1	Add or subtract	4 + 7 = 11
2	Multiply	4 × 7 = 28
3	Multiply, then add or subtract a small number	4 × 7 + 1 = 29
4	Square, or square and adjust	$4^2 + 1 = 17$
5	Digit tricks (sum, reverse)	see the digit lesson

Check every row

Rows: (3, 4, 13), (5, 2, 11), (6, 3, ?).

Try the sum. 3 + 4 = 7, not 13. Drop it.
Try the product. 3 × 4 = 12. The answer 13 is one more.
Guess: third = first × second + 1.
Test on row 2. 5 × 2 + 1 = 11. It fits.
Apply to row 3. 6 × 3 + 1 = 19.

Tip: Write the rule as a small formula, like c = a × b + 1. Then test that formula on the spare row.

Start from the biggest number

The biggest number in a row is usually the answer of the rule. The smaller numbers are the inputs.

Ask: how do the small numbers build the big one? 26 comes from 5 as $5^2 + 1$ . 21 comes from 4 and 5 as 4 × 5 + 1. Reading left to right hides the rule. Building the biggest number shows it.

Read the size of the answer

Compare the answer with the inputs. This tells you which family to try.

The answer is	Try first
About the sum of the inputs	Add or subtract
About the product	Multiply, then adjust
Close to a square number	Squares, like $a^2 \pm b$
Smaller than both inputs	Difference, average or a digit rule

Two grids and one hole

Often two complete grids sit beside a grid with a "?". The rule works inside each grid, row by row.

Find the rule in grid 1.
Test it on grid 2. It must fit every row.
Apply it to the row with the "?".

Grid 1 rows: (2, 3, 7) and (3, 4, 13). Rule: a × b + 1. Grid 2 has (5, 6, 31). Check: 5 × 6 + 1 = 31. So (4, 5, ?) gives 4 × 5 + 1 = 21.

Wrong options are planted

The examiner puts the most tempting wrong answer in the options. If the rule is a × b + 1, the plain product is there as bait.

For row (6, 3, ?), the plain product is 18. The right answer is 19. If your answer matches a simpler version of the rule, test that simple rule on a complete row. It will fail.

Watch: Never answer from one row. Two rows must agree.

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common3 practice Q

Third number is a multiple of the first

How to spot it:

Rows of two numbers. The second is a fixed multiple of the first, or a plain sum or difference.

Method

Divide the second number by the first in each row.
If the answer is the same, that is the multiplier.
Multiply the first number of the last row by it.

Why it works:

The simplest rules are tried first, and this one is the simplest.

Try this

Rows: 4 → 20, 6 → 30, 5 → ?. Find ?.

Show solution

20 ÷ 4 = 5 and 30 ÷ 6 = 5.
The rule is: second = first × 5.
5 × 5 = 25.

Answer

Type 2very common3 practice Q

Square of the number, then adjust

How to spot it:

The answers sit close to perfect squares like 10, 17, 26 or 48.

Method

Square the first number in each row.
Find what is left over: add or subtract a number.
The leftover must be the same in every row.
Apply to the last row.

Why it works:

A square with a small correction is a favourite exam rule.

Try this

Rows: 3 → 10, 4 → 17, 5 → ?. Find ?.

Show solution

$3^2 = 9$ , and 10 is 1 more.
$4^2 = 16$ , and 17 is 1 more.
Rule: $n^2 + 1$ . So $5^2 + 1 = 26$ .

Answer

Type 3common3 practice Q

Two grids, one rule

How to spot it:

Two complete grids and a third grid with one missing cell.

Method

Find the row rule in grid 1.
Test it on grid 2. It must fit.
Use it on the row with the ?.

Why it works:

The second grid is a free check of the rule.

Try this

Grid 1 rows: (2, 3, 7) and (3, 4, 13). Grid 2 rows: (4, 5, ?) and (5, 6, 31). Find ?.

Show solution

Grid 1: 2 × 3 + 1 = 7 and 3 × 4 + 1 = 13.
Check grid 2: 5 × 6 + 1 = 31. It fits.
4 × 5 + 1 = 21.

Answer

Type 4common3 practice Q

Two rules fit the first row

How to spot it:

The first row fits more than one rule. Only a second row can choose between them.

Method

Write every rule that fits row 1.
Test each one on row 2.
Keep the rule that fits both rows.
Use it on the last row.

Why it works:

Setters build rows so that a wrong rule survives one row but not two.

Try this

Rows: (2, 2, 4), (3, 5, 15), (6, 4, ?). Find ?.

Show solution

Row 1: both 2 + 2 and 2 × 2 give 4.
Row 2: 3 + 5 = 8, not 15. But 3 × 5 = 15.
The rule is a × b. So 6 × 4 = 24.

Answer

Type 5common

Cube of the number, then adjust

How to spot it:

The answers jump fast: 9, 28, 65. They sit next to cubes like 8, 27, 64.

Method

Cube the first number in each row.
Check the leftover. It must be the same in every row.
Apply the rule to the last row.

Why it works:

When answers grow faster than squares, a cube is the next family to try.

Try this

Rows: 2 → 9, 3 → 28, 4 → 65, 5 → ?. Find ?.

Show solution

$2^3 = 8$ , and 9 is 1 more.
$3^3 = 27$ and $4^3 = 64$ . Both are 1 less than the answer.
Rule: $n^3 + 1$ . So $5^3 + 1 = 126$ .

Answer

126

Type 6common

Multiply, then add a number

How to spot it:

Each answer is a little more than a multiple of the first number.

Method

Divide each answer by the first number to guess the multiplier.
The leftover gives the number to add.
Test on both rows, then apply.

Why it works:

Two small operations in a row are still a simple rule.

Try this

Rows: 5 → 11, 8 → 17, 12 → ?. Find ?.

Show solution

Try × 2: 5 × 2 = 10 (1 short), 8 × 2 = 16 (1 short).
Rule: first × 2 + 1.
12 × 2 + 1 = 25.

Answer

Formula sheet

Rule check

f(a_1, b_1) = c_1 \text{ and } f(a_2, b_2) = c_2 \Rightarrow \text{use } f

One row fitting proves nothing. Two rows fitting is strong proof.

Common families

a + b,\ a - b,\ ab,\ ab \pm k,\ (a + b)k,\ a^2 \pm b

Sums and products cover most puzzles. Try them first.

Shortcuts that save time

⚡ Test the spare row

Write your guessed rule as a formula. Test it on the second complete row. Only then use it on the row with the ?. This takes ten seconds and stops the most common wrong answer.

Example

Row 1: 3, 4, 13. Row 2: 5, 2, 11. Row 3: 6, 3, ?. Find ?.

Show solution

Guess c = a × b + 1. Row 1: 3 × 4 + 1 = 13.
Test row 2: 5 × 2 + 1 = 11. It fits.
Apply to row 3: 6 × 3 + 1 = 19.

Answer

⚡ Build the biggest number first

Pick the biggest number in the row. Try to make it from the smaller ones. This finds the rule faster than reading left to right.

Example

Row 1: 4, 7, 29. Row 2: 5, 3, 16. Row 3: 6, 2, ?. Find ?.

Show solution

Biggest is 29. 4 × 7 = 28, so 29 = 4 × 7 + 1.
Test row 2: 5 × 3 + 1 = 16. It fits.
Row 3: 6 × 2 + 1 = 13.

Answer

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Fitting a rule to the first row and answering at once.

Test the rule on the second complete row before you use it.

Mistake 02

Reading the figure only left to right.

The rule may run down a column or across opposite parts. Try those too.

Mistake 03

Using a three-step rule when a simple one exists.

Real exam rules are short. Try add, multiply and square first.

Mistake 04

Picking the option that equals the plain product or plain square.

That is often bait. Check that the plain rule really fits a complete row.

Mistake 05

Mixing the numbers of grid 1 with grid 2.

Find the rule in one grid. Then apply it on the other grid.

Quick revision

Read this the night before the exam.

Test rules in this order: add or subtract, multiply, multiply and adjust, squares, digit tricks.
The rule must fit every complete row. One fit is luck.
The biggest number in a row is usually the answer of the rule.
The size of the answer tells you the family: sum, product or square.
Two grids: find the rule in grid 1, test on grid 2, then apply it.
The plain product or plain square is often a planted wrong option.

Practice: 12 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 12 questions

Suggested time 8 min · wrong answers go to your mistake notebook automatically.

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