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Missing Number Problems: Undoing an Equation

Which inverse operation solves a missing number problem depends on where the blank sits. Worked examples, the two shapes that trip people up, and a check.

By Published Updated 6 min read

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Key takeaways

  • Every missing number problem is solved by applying the operation that undoes the one written in the equation, then checking the result by substituting it back.
  • Which inverse you need depends on where the blank sits: addition and multiplication commute, so the blank can be on either side, but subtraction and division do not.
  • When the blank is the amount being taken away, the answer is the starting number minus the result, not the two numbers added — and the same reversal applies to division.
  • Treating the equals sign as an instruction to compute rather than as a statement that two sides have the same value is the underlying error, and it predicts later difficulty with algebra.
  • A chain with more than one operation is peeled from the outside in: undo whatever was done last, first.
  • Substitution is a complete check. If the number you found makes the original equation true, it is the answer, whatever route you took to it.

A missing number problem hands you an equation with a gap in it — 47 + ? = 112 — and asks what belongs in the gap. One move solves all of them: apply the operation that undoes the one written in the equation. What makes them worth an article is that which inverse you need depends on where the gap sits, because subtraction and division do not behave like addition and multiplication. Get that wrong and the method fails quietly, producing a plausible number that is not the answer.

This is not a sequence puzzle. There is no pattern to spot and no list of terms to extend; the rule is already written down in front of you, and the work is entirely in reversing it. If you are here for patterns instead, the taxonomy of sequence types is in our guide to number sequence puzzles, and the rest of the logic and pattern puzzles hub covers the other families.

The equals sign is a statement, not an instruction

Most of the trouble starts before any arithmetic does. Read 47 + ? = 112 as "forty-seven plus something, now give me the answer" and the gap has nowhere to go. Read it as "the left side and the right side name the same number" and the gap becomes a constraint you can work with.

The distinction has a name in mathematics education — an operational versus a relational reading of the equals sign — and it is unusually consequential. A study by Percival Matthews and Lynn Fuchs, published in Child Development, tracked children's understanding of the equals sign in second grade against their algebra performance two years later, and reported that equal-sign knowledge was the strongest single predictor of later algebraic competence in their model, accounting for as much unique variance as the other predictors combined. Children holding the operational reading characteristically reject formats like 4 + 1 = 2 + 3 as wrong, because nothing after the equals sign looks like an answer.

That matters for puzzles, because good missing number problems use those layouts deliberately. 8 + 7 = ? + 6 is not a trick. The left side is 15, so the right side must also be 15, so the gap is 9. Check: 9 + 6 = 15.

Choosing the inverse depends on where the gap is

Addition and multiplication are commutative, so a gap on either side is handled the same way. Subtraction and division are not, and that is where the two common wrong answers come from. The table below gives the move for each of the seven shapes you will actually meet, with a, b and c standing for numbers that are written down.

Equation shape The move Note
a + ? = c c - a Order does not matter
? + b = c c - b Same move as above
? - b = c c + b The gap is the starting amount
a - ? = c a - c Not a + c
a * ? = c c / a Order does not matter
? / b = c c * b The gap is the amount being divided
a / ? = c a / c Not a * c

The two bolded rows are the whole difficulty. When the gap is the amount being taken away or divided by, it sits in the position the inverse operation also wants, and the operation appears to reverse itself. Worked through, they are unremarkable:

  • 91 - ? = 57. The gap is 91 - 57 = 34. Check: 91 - 34 = 57.
  • 204 / ? = 17. The gap is 204 / 17 = 12. Check: 204 / 12 = 17, and 17 x 12 = 204.

Compare those with the friendlier versions, where the gap is the starting amount:

  • ? - 38 = 26. The gap is 26 + 38 = 64. Check: 64 - 38 = 26.
  • ? / 9 = 13. The gap is 13 x 9 = 117. Check: 117 / 9 = 13.

A quick sanity test separates the two cases without memorising the table. Ask whether the gap should be bigger or smaller than the number you can see. In 91 - ? = 57 you are removing something from 91 and landing on 57, so the gap has to be smaller than 91 — which rules out 91 + 57 before you calculate anything.

Chains: undo the last thing first

Once an equation contains two operations, a single inverse is not enough. Peel it from the outside in, undoing whichever operation was applied last.

6 * ? + 9 = 51

Nine was added at the end, so remove it first: 51 - 9 = 42. That leaves 6 * ? = 42, and dividing gives 7. Substituting back: 6 x 7 = 42, and 42 + 9 = 51.

A longer one, (? + 5) * 4 - 12 = 40:

  1. Undo the subtraction: 40 + 12 = 52.
  2. Undo the multiplication: 52 / 4 = 13.
  3. Undo the addition: 13 - 5 = 8.

Check forwards: 8 + 5 = 13; 13 x 4 = 52; 52 - 12 = 40. That order reversal is the part readers get wrong often enough to deserve its own treatment, which is why working backwards has a separate article covering longer chains and the problems where the chain is buried in a story.

Missing operators, and why some have two answers

A related family gives you all the numbers and removes the symbol instead: 12 ? 4 ? 3 = 9. The method changes, because you are now searching a small space rather than inverting a known step — and the space really is small, since there are only four operators to try in each gap. Reading left to right, 12 - 4 + 3 = 11, 12 / 4 + 3 = 6, and 12 x 4 / 3 = 16. The combination that lands on 9 is 12 / 4 x 3. Check: 12 / 4 = 3, and 3 x 3 = 9.

NRICH, the mathematics education project based in the Faculty of Mathematics at the University of Cambridge, uses this format in its "Which Symbol?" task, and its teachers' notes point out that several of the two-gap versions have two valid solutions rather than one. That is worth knowing before you decide you have the wrong answer: a missing-operator puzzle is only guaranteed unique if the setter checked that it was.

Practising the move rather than the puzzle

Reading a method and owning it are different things, and the gap between them closes through repetition on items you have not seen before. IQ Test & Brain Training Games, which is our own app, ships a Missing Number Puzzle mode built around this exact task, with Easy, Medium and Hard settings on a 75-second clock, plus an Equation Builder mode that works the other way round — you assemble the pieces that make a given equation true. The point of the timer is not speed for its own sake; it is that a time limit discourages solving by trial and error, which is the habit that stops the inverse from becoming automatic.

Two cautions about drilling this. First, accuracy before speed: an inverted sign you never notice is worse than a slow correct answer, and the quickest way to install the error permanently is to practise it at pace. Second, always substitute. Substitution is a complete check for a single unknown — if your number makes the original equation true, it is right, regardless of how you found it.

The arithmetic underneath is ordinary, and if the division steps are the slow part rather than the logic, the methods in our guide to mental division and to multiplication shortcuts will do more for your times than more puzzle practice will. You can see how the app works and how progress is tracked on the IQ Test & Brain Training Games app page, and the rest of our writing on arithmetic and reasoning sits in the education and brain training section.

The CEMC at the University of Waterloo publishes free courseware and a weekly problem set covering this material from upper primary through senior secondary, which is a good next step if you want problems that are graded rather than generated.

Good to know

Frequently asked questions

What is the inverse operation of subtraction?
Addition, but only when the blank is the number being reduced. In an equation shaped like blank minus 38 equals 26, the blank is the starting amount, so you add: 26 + 38 = 64. In 91 minus blank equals 57, the blank is the amount taken away, and adding gives nonsense. There you subtract the result from the starting number: 91 - 57 = 34. The operation written in the equation does not by itself tell you which move to make; the position of the blank does.
How do I solve a missing number problem with two operations?
Undo the operations in reverse order, starting with whichever was applied last. In six times a blank, plus nine, equals 51, the addition of 9 happened last, so remove it first: 51 - 9 = 42. That leaves six times the blank equals 42, and dividing gives 7. Substituting back confirms it: 6 times 7 is 42, plus 9 is 51. Working from the outside in keeps each step to a single inverse.
Why do some missing number problems have the blank on the right of the equals sign?
Because the equals sign means the two sides have the same value, not that an answer follows. An equation reading 8 + 7 = blank + 6 is perfectly ordinary: the left side is 15, so the blank must make the right side 15 as well, which means 9. Non-standard layouts like this are used deliberately in teaching, because learners who read the equals sign as an instruction to compute tend to answer 15 instead.
Is a missing number puzzle the same as algebra?
It is algebra with a blank in place of a letter. Replacing the box with x turns a blank times 12 equals 156 into 12x = 156, and the solving move is identical. The main difference is presentation: a puzzle usually guarantees a whole-number answer and a single unknown, so the arithmetic stays manageable and the focus stays on choosing the right inverse.

Sources

  1. Which Symbol? (opens in a new tab)NRICH, University of Cambridge — accessed
  2. Keys to the Gate? Equal Sign Knowledge at Second Grade Predicts Fourth-Grade Algebra Competence (opens in a new tab)Child Development, via PubMed Central — accessed
  3. CEMC Courseware (opens in a new tab)Centre for Education in Mathematics and Computing, University of Waterloo — accessed

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