11+ · Topic guide

Missing Digit and Calculation Puzzles

Missing digit and calculation puzzles hide one or more digits of a written addition, subtraction or multiplication behind a letter or blank, and ask a student to work out the hidden digit or digits using place value and known column-arithmetic facts such as carrying and borrowing, rather than by guessing and checking every possibility.

Year 5-6 (11+)Maths and Numerical ReasoningGL AssessmentCEM

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Before you start

Make sure you're comfortable with these topics first:

Working Backwards with Inverse Operations Place Value and Ordering

Method

Start with the units (rightmost) column, since it is the only column that does not depend on a carry from an earlier column.
Remember that a column total of 10 or more means a digit is carried into the next column to the left.
When a digit stands for an unknown, work out every value it could take that fits the column you are looking at, then use a later column to check which value is actually correct.
In a multiplication puzzle, it is often quicker to divide the known total by the known factor to find the missing digits directly, then check the answer using the individual columns.
If the same letter appears more than once in a puzzle, it must represent the same digit every time.
Once you have found every missing digit, rewrite the full calculation with real numbers and check it is correct.

Worked example

In the addition below, A and B represent single digits. 3A7 + 28B ----- 625 Find the values of A and B, showing your working.

Add the units column: 7 + B must end in 5. Since 7 + B is at least 7, the only way to end in 5 is 7 + B = 15, so B = 8, carrying 1 to the tens column.
Add the tens column, including the carry: A + 8 + 1 = A + 9 must end in 2. Since A is a single digit, A + 9 = 12, giving A = 3 and carrying 1 to the hundreds column.
Check the hundreds column: 3 + 2 + 1 (carried) = 6, which matches the given hundreds digit.
So A = 3 and B = 8.
Verify by working the full sum: 337 + 288 = 625, which matches the original puzzle.

Practice questions

Type your answer and press Check to be marked straight away, or reveal the answer and mark yourself.

Q1Show answer

Answer: 4 (44 + 27 = 71)

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Q294 - 3Y = 58, where Y is the missing units digit of the two-digit number 3Y. Find Y.Show answer

Answer: 6 (94 - 36 = 58)

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Q36 x Z = 42, where Z is a single digit. Find Z.Show answer

Answer: 7

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Q4AB + 15 = 62, where AB is a two-digit number with tens digit A and units digit B. Find the two-digit number AB.Show answer

Answer: 47 (47 + 15 = 62)

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Q5Show answer

Answer: 8

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Q6Show answer

Answer: 2 (90 - 52 = 38)

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Q7Show answer

Answer: 4 (9 x 6 = 54)

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Q8Show answer

Answer: 3 (100 - 63 = 37)

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Exam-style questions

Written in the style of a 11+ exam paper, with a full mark scheme.

Q1[4 marks]

In the addition below, A and B represent single digits. 5A6 + 37B ----- 934 Find the values of A and B, showing your working.

Show mark scheme

Tick each line you got. Your score builds from the marks on the scheme.

units column, 6 + B must end in 4; since 6 + B is at least 6, 6 + B = 14, so B = 8, carrying 1.1 mark
tens column, A + 7 + 1 = A + 8 must end in 3; A + 8 = 13, so A = 5, carrying 1.1 mark
hundreds column check, 5 + 3 + 1 (carried) = 9, matching the given hundreds digit.1 mark
correct final values, A = 5 and B = 8 (verified: 556 + 378 = 934).1 mark

Nothing ticked yet - 4 available

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Q2[5 marks]

In the multiplication below, P and Q represent single digits. PQ x 4 ---- 296 Find the values of P and Q, showing your method.

Show mark scheme

Tick each line you got. Your score builds from the marks on the scheme.

recognise that PQ x 4 = 296, so PQ = 296 / 4.1 mark
divide, 296 / 4 = 74, so PQ = 74 (280 / 4 = 70, then 16 / 4 = 4, giving 70 + 4 = 74).1 mark
so P = 7 and Q = 4.1 mark
check the units column directly: Q x 4 must end in 6; since Q = 4, 4 x 4 = 16, which ends in 6, confirming Q = 4 with a carry of 1 to the tens.1 mark
check the tens column: (P x 4) + 1 (carried) = (7 x 4) + 1 = 29, giving hundreds digit 2 and tens digit 9, which matches 296.1 mark

Nothing ticked yet - 5 available

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Free printable worksheet

Want more practice on paper? Download the missing digit and calculation puzzles worksheet pack - 8 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.

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This topic is chapter 27 of 11+ Maths Workbook 4, the whole course as one free printable PDF.

Next topics

Divisibility Rules and Number Property Puzzles Worded Problems Set F

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