Concept 2 of 5
Work-Backwards Puzzles
Today's big idea
Start at the end and undo each step with its opposite, last step first.
Spending undoes with adding, and doubling undoes with halving.
How it works
👀 What's going on?
Some puzzles hand you the ending and ask for the beginning. Reading them forwards gets you nowhere, because the number you would start from is the very thing being asked for.
💡 Here's the trick
Look at the chain above. Working from the finish, each step is undone by its opposite: money spent is added back, money received is taken off, a doubling is halved. Undo the last thing that happened first, then work back to the start.
🪄 How to do it
- 1.Start at the finishing amount shown in the chain above.
- 2.Take the last step that happened and do its opposite.
- 3.Keep undoing backwards until you reach the beginning.
✍️ Try it
A number is doubled to make 18, so what was it? Then: Ravi gave away 12 marbles, then found 5, and ended with 20.
Halving undoes doubling, so the number was 9. For Ravi, take off the 5 he found and add back the 12 he gave away. That is 20 minus 5, then plus 12, giving 27 marbles.
⚠️ Watch out
You might be thinking you undo the steps in the order the story tells them. You do not. Ravi's finding came last, so it is undone first. Take the 12 off before the 5 and the answer comes out wrong by seventeen.
🌟 Why this matters
Working out what you started the week with, or how much was in an account before a run of spending, is this same walk backwards. The ending is often the only figure you actually know.
Exam tip: Run your answer forwards through the story to check; 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 is doubled, then 5 is added, giving 25. What was the starting number?”
- 1
Undo the last step
Last step was +5, so subtract: 25 − 5 = 20. ⏪
- 2
Undo the doubling
Doubling undoes by halving: 20 ÷ 2 = 10.
- 3
Answer ✓
Start = 10
Now you try one — same steps, different story 👇
See it in action
⚡ Time to play!
Undo each step backwards: spent → add back, doubled → halve.
REWIND · ROUND 1 OF 3
After spending ₹40, Anu has ₹25
How much did she start with?
Quick check
✅ VERIFICATION
Which move undoes 'divided by 4'?
⚡ Finish the activity above first to answer.
Keep going!
Up next: Combine-the-Groups Sums
Add the groups for a total, subtract to compare.