Missing Number

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Digit-sum and digit-reversal rules

⏱ 5 min read🧩 7 question types🎯 12 practice Q

The idea in one minute

Some puzzles work on the digits of the numbers, not on their values. The rule may add the digits of each number, reverse the digits, or multiply the digits. Try a digit rule when ordinary sums and products do not fit.

When the rule uses digits

Some rows look odd because the values do not match: 28, 9, 21. Adding the digits explains it: 2 + 8 = 10, then 10 + 9 + 2 = 21.

Three digit tools cover most puzzles. Learn their short names.

Tool	Meaning	Example
$S(n)$	sum of the digits	$S(47) = 4 + 7 = 11$
$R(n)$	digits reversed	$R(43) = 34$
$P(n)$	product of the digits	$P(47) = 4 \times 7 = 28$

Digit-sum rules

The answer is small, but the numbers are two-digit. Try these:

$S(a) + S(b)$ : 47 and 38 give 11 + 11 = 22.
$S(a + b)$ : add first, then sum the digits. 68 gives 14.
A digit sum times a number, like $S(a) \times b$ .

Test one variant on a complete row. Then test it on the next row. Only one variant fits all.

Reversal rules

Look for mirror pairs, like 43 and 34.

$R(a) + R(b)$ : reverse both numbers, then add. 34 + 41 = 75.
$R(a + b)$ : add first, then reverse the sum. 39 becomes 93.

Tip: Reversing first and adding first can give different answers. Test both on a row where the digits carry over.

Digit-product rules

The answer is very small. Multiply the digits of each number.

$P(24) = 2 \times 4 = 8$ and $P(35) = 15$ . So $P(47) = 28$ .

Watch: A number with a 0, like 40 or 10, has a digit product of 0. This usually rules out a product rule at once.

A full walk-through

Rows: (39, 25, 10), (57, 18, 12), (68, 26, ?).

Plain sum: 39 + 25 = 64, not 10. Drop it.
Digit sums added: 3 + 9 = 12 and 2 + 5 = 7. Total 19, not 10. Drop it.
Add first, then take the digit sum: 64 gives 6 + 4 = 10. It fits.
Test row 2: 57 + 18 = 75, and 7 + 5 = 12. It fits.
Apply: 68 + 26 = 94, and 9 + 4 = 13.

How to spot a digit rule

The answers are far too small or too big for plain sums and products.
The answers are near 5 to 30 while the numbers are two-digit.
The same digits appear again and again: 43 and 34, or 62 and 26.

A quick check with 9

A number and its digit sum leave the same remainder when divided by 9. This helps to check a digit-sum answer. For example 47 + 38 = 85, and $S(47) + S(38) = 22$ . Both 85 and 22 leave remainder 4.

Add a fixed number

Sometimes the digit sums are close but always short by the same amount. Rows (28, 9, 21) and (36, 23, 16) show it. The digit sums total 19 and 14. The answers are 2 more each time. So the rule is $S(a) + S(b) + 2$ .

Reversal with a zero

Reversing the inputs when the rule reverses the sum, or the other way round.
A sum that ends in 0, like 40. Reversal gives 04 = 4. Setters avoid this. If you see it, you may have the wrong rule.

Rule: Use the same variant that fitted the complete rows. Do not switch to another one for the last row.

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

Sum of the digit sums

How to spot it:

The answers are small (5 to 30) while the numbers are two-digit.

Method

Find S(a) and S(b) for each row.
Add them and compare with the third number.
Check every row, then apply.

Why it works:

Digit sums shrink big numbers to answer-sized ones.

Try this

Rows: (36, 24, 15), (28, 19, 20), (47, 38, ?). Find ?.

Show solution

36: 3 + 6 = 9. 24: 2 + 4 = 6. Total 15.
28: 10. 19: 10. Total 20.
47: 11. 38: 11. Total 22.

Answer

Type 2very common3 practice Q

Reverse each number, then add

How to spot it:

Digits mirror across the row, like 43 beside 34.

Method

Reverse both numbers.
Add the two reversed numbers.
Confirm on every row, then apply.

Why it works:

Reversal is the most reused digit trick.

Try this

Rows: (23, 14, 73), (46, 25, 116), (58, 17, ?). Find ?.

Show solution

23 → 32 and 14 → 41. Sum 73.
46 → 64 and 25 → 52. Sum 116.
58 → 85 and 17 → 71. Sum 156.

Answer

156

Type 3common2 practice Q

Product of the digits

How to spot it:

The answers are tiny, even smaller than the digit sums.

Method

Multiply the digits of the first number.
Compare with the answer.
Check every row, then apply.

Why it works:

Digit products shrink numbers even more than digit sums.

Try this

Rows: 24 → 8, 35 → 15, 47 → ?. Find ?.

Show solution

2 × 4 = 8.
3 × 5 = 15.
4 × 7 = 28.

Answer

Type 4common2 practice Q

Digit sum times the other number

How to spot it:

One number is shrunk by its digit sum. The other stays whole.

Method

Find the digit sum of the first number.
Multiply it by the second number.
Check every row, then apply.

Why it works:

Mixing a digit tool with a plain number hides the rule one level deeper.

Try this

Rows: (24, 3, 18), (35, 2, 16), (47, 2, ?). Find ?.

Show solution

24: 6, and 6 × 3 = 18.
35: 8, and 8 × 2 = 16.
47: 11, and 11 × 2 = 22.

Answer

Type 5common

Add first, then reverse the sum

How to spot it:

The answers are mirror images of the row sums, like sum 42 and answer 24.

Method

Add the first two numbers.
Reverse the digits of the sum.
Confirm on a second row. Then apply.

Why it works:

The reverse is applied to the result, not to the inputs.

Try this

Rows: (24, 18, 24), (32, 13, 54), (26, 15, ?). Find ?.

Show solution

24 + 18 = 42. Reversed 24.
32 + 13 = 45. Reversed 54.
26 + 15 = 41. Reversed 14.

Answer

Type 6common

Digit sum of the total

How to spot it:

Adding the digit sums separately does not fit, but the digit sum of the total does.

Method

Add the two numbers first.
Then find the digit sum of the total.
Check S(a) + S(b) too. It differs when there is a carry.

Why it works:

Adding before or after taking digit sums gives different answers when digits carry over.

Try this

Rows: (39, 25, 10), (57, 18, 12), (68, 26, ?). Find ?.

Show solution

39 + 25 = 64. 6 + 4 = 10.
57 + 18 = 75. 7 + 5 = 12.
68 + 26 = 94. 9 + 4 = 13.

Answer

Type 7occasional

Digit sums plus a fixed number

How to spot it:

The digit sums are close to the answer but a small fixed amount is left over.

Method

Add the digit sums of both numbers.
Find the leftover to reach the answer.
The leftover must be the same in every row.
Apply to the last row.

Why it works:

A fixed add-on is a small extra step on the digit-sum rule.

Try this

Rows: (28, 9, 21), (36, 23, 16), (45, 17, ?). Find ?.

Show solution

28: 10 and 9. Total 19. 21 is 2 more.
36: 9 and 23: 5. Total 14. 16 is 2 more.
45: 9 and 17: 8. Total 17. Add 2 to get 19.

Answer

Formula sheet

Digit sum

S(47) = 4 + 7 = 11

Add the digits of the number.

Reversal

R(43) = 34

Write the digits in the opposite order.

Digit product

P(47) = 4 \times 7 = 28

Multiply the digits of the number.

Shortcuts that save time

⚡ Switch to digits when standard rules fail

Give ordinary rules about 30 seconds. If both complete rows refuse them, test these in order: digit sums, reverse of the sum, digit sums plus a number.

Example

Row 1: 28, 9, 21. Row 2: 36, 23, 16. Row 3: 45, 17, ?. Find ?.

Show solution

Sums and products fail.
Row 1: 10 + 9 = 19, and 21 is 2 more. Row 2: 9 + 5 = 14, and 16 is 2 more.
Row 3: 9 + 8 + 2 = 19.

Answer

⚡ Add first, then flip

A reversal answer must read backwards cleanly. Add the two numbers first, then reverse the sum. Confirm on two rows.

Example

Row 1: 24, 18, 24. Row 2: 32, 13, 54. Row 3: 26, 15, ?. Find ?.

Show solution

24 + 18 = 42. Reversed: 24.
32 + 13 = 45. Reversed: 54.
26 + 15 = 41. Reversed: 14.

Answer

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Reversing the inputs when the rule reverses the sum.

Test both orders on a complete row. Use the one that fits.

Mistake 02

Slipping on digit sums of numbers with 0 or repeated digits.

Write each digit sum in a short line, like 40 → 4 + 0 = 4.

Mistake 03

Skipping the second-row check because the rule looks digital.

Check every complete row anyway.

Mistake 04

Adding the digits of the numbers when the rule adds first, then sums the digits.

Test S(a) + S(b) and S(a + b). They differ when there is a carry.

Mistake 05

Changing to another variant for the last row.

Use the same variant that fitted the complete rows.

Quick revision

Read this the night before the exam.

S(n) is the digit sum, R(n) reverses the digits, P(n) multiplies the digits.
Common rules: S(a) + S(b), S(a + b), R(a) + R(b), R(a + b), P(a), S(a) × b.
Test each variant on a complete row. Only one fits all the rows.
Answers far smaller than the numbers point to a digit rule.
Mirror pairs like 43 and 34 point to reversal.
A 0 in a number kills a digit-product rule.

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 9 min · wrong answers go to your mistake notebook automatically.

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