Working Backwards

1. 1The Starting PileHere is Mia's pile at the very start — a full stack. We do not know how many books it holds yet, so it wears a ? mark.Start: ? books
2. 2Dividing by 2Mia splits the pile into 2 equal parts and keeps just one part. The other parts leave the pile.? ÷ 2
3. 3Giving 3 Books to a FriendMia takes the top 3 books of the pile and gives them to her friend. The pile drops by exactly 3 books.- 3 books
4. 4Buying 5 More BooksFinally, Mia buys 5 new books and puts them on the pile. Now we know for sure: the pile holds 12 books.+ 5 → 12 books
5. 5Working BackwardsKnow the end, find the start: travel backwards, undo the operations. We start from 12 books and travel back station by station, undoing each event in reverse order.12 → ?
6. 6Undo the BuyingTake the 5 bought books back off the pile: 12 - 5 = 7. Before buying, Mia had 7 books.12 - 5 = 7
7. 7Undo the GivingHer friend gives the 3 books back and the pile grows: 7 + 3 = 10. Before giving any away, Mia had 10 books.7 + 3 = 10
8. 8Undo the DivisionThe divided parts come back to the pile: 10 × 2 = 20. The pile stands as tall as at the start.10 × 2 = 20
9. 9The ? Becomes a NumberThe ? at the first station turns into 20. Mia started with 20 books.Start: 20 books
10. 10Check: Run It ForwardReplay the story quickly from 20 books: divide by 2, give 3 to a friend, buy 5 more — and land exactly on 12 books.20 ÷ 2 = 10 → 10 - 3 = 7 → 7 + 5 = 12
11. 11Wrap-Up: Inverse PairsEvery operation has an inverse: dividing pairs with multiplying, giving away with getting back, buying with subtracting. When a problem tells you the end, travel backwards and undo step by step.Divide ↔ Multiply · Add ↔ Subtract

Change the numbers to test the engine with any sum & difference problem: