Concept 5 of 7
Missing Numbers
Today's big idea
Start at the answer and undo each step backwards, swapping every operation for its opposite.
Running the number forward again is how you check you undid it correctly.
How it works
👀 What's going on?
A machine swallows a number, does something to it, and spits a new one out. Usually you are told what went in. Sometimes you are told what came out and asked what went in, and that feels like being handed the end of a story first.
💡 Here's the trick
Look at the two rows above. Going forward the machine doubles and then adds 3, and 11 comes out. To find what went in, walk it backwards from 11 and swap every step for its opposite: take away 3, then halve. Undo the last thing first, exactly as you undo shoes before socks.
🪄 How to do it
- 1.Start at the answer and take the last step of the machine above, not the first.
- 2.Swap that step for its opposite: plus undoes with minus, times undoes with divide.
- 3.Keep going back until you reach the start, then run it forward to check.
✍️ Try it
A number goes in, is multiplied by 4, then has 5 taken off, and 19 comes out. What went in? Then find the missing digit in 3_ + 24 = 61.
Undo the last step first: 19 + 5 = 24, then 24 divided by 4 is 6. Check it forward and 6 × 4 = 24, then 24 − 5 = 19. For the digit, 61 − 24 = 37, so the hidden digit is 7.
⚠️ Watch out
You might be thinking you should undo the steps in the order you read them. Undo the multiply first on that puzzle and you get 19 divided by 4, which is not even whole. Backwards means backwards: last step first, every time.
🌟 Why this matters
Working out how much more money you need, or how many points behind you are, is finding a missing number. You know the finish and you are hunting the start.
Exam tip: Always run your answer forward through the machine; it takes seconds and catches a flipped step.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“A number goes into a machine → ×2 → +3 → out 11. What number went in?”
- 1
Start at the output
The machine ended at 11. 🎰
- 2
Undo the last step
Last step was +3, so subtract: 11 − 3 = 8.
- 3
Undo the first step
First step was ×2, so divide: 8 ÷ 2 = 4.
- 4
Answer ✓
Went in = 4
Now you try one — same steps, different story 👇
See it in action
⚡ Time to play!
Tap ▲ / ▼ to change the number in the box. Make both sides of the = match!
MISSING NUMBER · ROUND 1 OF 3
11 < 14
Too small — tap ▲ to make the box bigger.
Quick check
✅ VERIFICATION
A number goes into a machine → ×3 → −4 → out 20. What number went in?
⚡ Finish the activity above first to answer.
Keep going!
Up next: Estimating Sums & Differences
296 + 405 is about 300 + 400 = 700.