Missing Number

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Grids with row (or column) rules

⏱ 5 min read🧩 7 question types🎯 21 practice Q

The idea in one minute

In a 3×3 grid, each row usually turns its first two numbers into the third with the same rule. Sometimes the rule runs down the columns instead. Find the rule on the complete rows, test it on every complete row, then fill the hole.

What a grid puzzle looks like

A grid has three rows of three numbers. One cell is a "?". Each row follows the same rule.

Row (4, 6, 10): here 4 + 6 = 10. The first two numbers make the third. Your job is to find this link fast.

Hunt for the link

Cover the third column. Ask how the first two numbers make the third. Try these in order.

Add or subtract: $a + b$ , $a - b$ .
Multiply: $ab$ . Then $2ab$ , $ab + k$ , $ab - k$ .
Squares: $a^2 + b$ , $a^2 - b$ , $a^2 + b^2$ .
Square of a sum: $(a + b)^2$ .
Sum times a number: $(a + b) \times k$ .
Average: $\dfrac{a + b}{2}$ .

Tip: If the third number is a perfect square, like 25, 49 or 144, try a square rule first.

Check every row

Rows: (4, 6, 10), (5, 3, 22), (7, 2, ?).

Sum: 4 + 6 = 10. It fits row 1. But 5 + 3 = 8, not 22. Drop it.
Product: 4 × 6 = 24, not 10. Drop it.
Try $a^2 - b$ . Row 1: 16 − 6 = 10. Row 2: 25 − 3 = 22. Both fit.
Apply: 49 − 2 = 47.

Watch: Stopping at row 1 would give 7 + 2 = 9. That is the planted wrong option. Always test the second row.

When rows fail, try columns

If no row rule fits, read the grid downwards. The third row may come from the first two rows, cell by cell.

Rows: (2, 3, 5), (4, 5, 6), (8, 15, ?). Check the columns: 2 × 4 = 8 and 3 × 5 = 15. So the third row is first row × second row. Then ? = 5 × 6 = 30.

The middle cell can also be the average of the outer cells. In (10, 14, 18) and (22, 25, 28), the middle number is the average. So (30, ?, 40) gives 35.

Sum, then a multiplier

Sometimes the third number is a multiple of the sum. Add the first two numbers. Then divide the third number by that sum.

Rows: (4, 7, 22) and (6, 3, 18). Row 1: 4 + 7 = 11, and 22 ÷ 11 = 2. Row 2: 6 + 3 = 9, and 18 ÷ 9 = 2. The multiplier is 2 in both rows. So (9, 5, ?) gives 14 × 2 = 28.

Circles and triangles

Match parts by position.

Circle: opposite parts often share a total. Pairs 12–8 and 15–5 both add to 20. So 18 pairs with 2.
Triangle: the centre may be the sum or product of the corners. Corners 2, 3, 4 give centre 24. Corners 1, 5, 3 give 15. So corners 2, 6, 3 give 36.

Write each matched set as a small sum. Look for the same link in every set.

Two complete grids

The rule stays the same in both grids. Only the numbers change. Solve the left grid, check it on the middle grid, then use it on the last grid.

Speed helps

Learn squares up to 25.
Use $a^2 - b^2 = (a + b)(a - b)$ to skip big squares: $7^2 - 3^2 = 10 \times 4 = 40$ .
Build the biggest cell first: $22 = (4 + 7) \times 2$ .

Rule: One rule, every row, then answer.

Question types you will see

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

Type 1very common2 practice Q

Row sums and differences

How to spot it:

The third number is close to the sum or difference of the first two.

Method

Test c = a + b on every row.
If it fails, test the difference a − b.
Use the rule that fits every row.

Why it works:

Adding is the most common way to build a grid.

Try this

Rows: (5, 8, 13), (6, 9, 15), (7, 11, ?). Find ?.

Show solution

5 + 8 = 13 and 6 + 9 = 15.
The rule is a + b.
7 + 11 = 18.

Answer

Type 2common2 practice Q

Squares added together

How to spot it:

The third number is much bigger than the first two, and it is a perfect square.

c = a^2 + b^2

Method

Square both numbers.
Add the squares.
Check every row, then apply.

Why it works:

Square rules make the third number too big for a plain sum or product.

Try this

Rows: (3, 4, 25), (6, 8, 100), (5, 12, ?). Find ?.

Show solution

$3^2 + 4^2 = 9 + 16 = 25$ .
$6^2 + 8^2 = 36 + 64 = 100$ .
$5^2 + 12^2 = 25 + 144 = 169$ .

Answer

169

Type 3common2 practice Q

Column rules

How to spot it:

No row rule fits. The grid may work downwards.

Method

Test third row = first row + second row, cell by cell.
Then test the product of the two rows.
Confirm on every complete column, then fill the hole.

Why it works:

The setter sometimes turns the rule to run down the columns.

Try this

Rows: (2, 3, 5), (4, 5, 6), (8, 15, ?). Find ?.

Show solution

Column 1: 2 × 4 = 8. Column 2: 3 × 5 = 15.
The third row is first row × second row.
Column 3: 5 × 6 = 30.

Answer

Type 4common2 practice Q

Circle and triangle figures

How to spot it:

Numbers sit around a circle, or at the corners of a triangle with a centre.

Method

Pair opposite parts of the circle. Test a fixed sum, then a fixed product.
For a triangle, test centre = sum or product of the corners.
Confirm on every matching set.

Why it works:

These figures repeat one link across matched parts.

Try this

A circle has opposite pairs 12 and 8, 15 and 5, and 18 and ?. Find ?.

Show solution

12 + 8 = 20 and 15 + 5 = 20.
Each opposite pair adds to 20.
20 − 18 = 2.

Answer

Type 5very common

Multiply, then add or subtract a number

How to spot it:

The third number is a little more or less than the product of the first two.

c = ab \pm k

Method

Multiply the first two numbers in each row.
Find what is left over. It must be the same in every row.
Apply to the last row.

Why it works:

A small change on a product is the most used disguise for multiplying.

Try this

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

Show solution

2 × 5 = 10, and 11 is 1 more.
4 × 3 = 12, and 13 is 1 more.
Rule: a × b + 1. So 6 × 4 + 1 = 25.

Answer

Type 6common

Sum, then multiply by a number

How to spot it:

The third number is even, or a multiple of a fixed number, and bigger than the sum.

c = (a + b) \times k

Method

Add the first two numbers.
Divide the third number by that sum.
The quotient k must be the same in every row.
Apply to the last row.

Why it works:

Adding first and then scaling is a common two-step rule.

Try this

Rows: (4, 7, 22), (6, 3, 18), (9, 5, ?). Find ?.

Show solution

4 + 7 = 11 and 22 ÷ 11 = 2.
6 + 3 = 9 and 18 ÷ 9 = 2.
(9 + 5) × 2 = 28.

Answer

Type 7common

Square of the first, minus the second

How to spot it:

The third number is close to the square of the first number.

c = a^2 - b

Method

Square the first number.
Subtract the third number from the square. The result should be the second number.
Check every row, then apply.

Why it works:

Squares with a subtraction often hide behind large third numbers.

Try this

Rows: (8, 15, 49), (7, 10, 39), (9, 6, ?). Find ?.

Show solution

$8^2 - 15 = 64 - 15 = 49$ .
$7^2 - 10 = 49 - 10 = 39$ .
$9^2 - 6 = 81 - 6 = 75$ .

Answer

Formula sheet

Sum times a number

c = (a + b) \times k

Rows (4, 7, 22) and (6, 3, 18) give k = 2.

Square minus the second

c = a^2 - b

Rows (8, 15, 49) and (7, 10, 39) fit.

Sum of two squares

c = a^2 + b^2

Rows (3, 4, 25) and (6, 8, 100) fit.

Average of two

c = \dfrac{a + b}{2}

Rows (12, 8, 10) and (20, 6, 13) fit.

Shortcuts that save time

⚡ Build the biggest cell first

Write the biggest cell of a row. Rebuild it from the other two. This shows the rule faster than reading left to right.

Example

Row 1: 4, 7, 22. Row 2: 6, 3, 18. Row 3: 9, 5, ?. Find ?.

Show solution

22 = (4 + 7) × 2.
Test row 2: (6 + 3) × 2 = 18. It fits.
Row 3: (9 + 5) × 2 = 28.

Answer

⚡ Copy the grid as three rows

Write the grid as three lines in your rough space, with the ? in place. Reading numbers off the figure while you calculate leads to mistakes.

Example

Row 1: 12, 8, 10. Row 2: 20, 6, 13. Row 3: 14, 4, ?. Find ?.

Show solution

10 = (12 + 8) ÷ 2.
Test row 2: (20 + 6) ÷ 2 = 13. It fits.
Row 3: (14 + 4) ÷ 2 = 9.

Answer

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Using grid 1's row rule on grid 2's columns.

Use the same direction in every grid. Rows with rows, columns with columns.

Mistake 02

Picking between two half-fitting rules by feel.

Test both rules on every complete row. Keep only the one that fits all.

Mistake 03

Squaring wrongly, like 13² = 196.

13² = 169. Learn squares up to 25.

Mistake 04

Stopping when row 1 fits.

Row 2 must fit too. A rule that fits one row is luck.

Mistake 05

Reading a column rule as a row rule.

If no row rule fits, read the grid downwards.

Quick revision

Read this the night before the exam.

Try in order: sum or difference, product, squares, (a + b)², sum times a number, average.
The rule must fit every complete row, and the second grid too.
Rows fail: try columns. Circles: opposite parts. Triangles: centre and corners.
Third numbers that are perfect squares point to a square rule.
Build the biggest cell from the others.
Learn squares up to 25. Use a² − b² = (a + b)(a − b).

Practice: 21 questions

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

Topic test · 10 questions

Suggested time 7 min · wrong answers go to your mistake notebook automatically.

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