Concept 10 of 10
Find the Rule
Today's big idea
From In/Out pairs, work out the single × or ÷ rule, then use it to predict any output.
4→28 and 5→35 both fit × 7, so that is the rule.
How it works
👀 What's going on?
A function machine takes a number, does something hidden to it, and returns another. Given a table of numbers going in and coming out, a mathematician reverse-engineers the rule inside — and then predicts every other answer.
💡 Here's the trick
Look at the In/Out table above. 4 becomes 28, 5 becomes 35, 6 becomes 42. Each out is its in multiplied by 7, so the hidden rule is × 7. Find the rule that fits every pair, and the machine holds no more secrets.
🪄 How to do it
- 1.Compare each in with its out and ask what single step connects them.
- 2.Try the times tables and the divide-bys: is every out the in times the same number, or divided by it?
- 3.Confirm the rule on a second and third pair before trusting it, then apply it.
✍️ Try it
In 48 → Out 6, In 56 → Out 7, In 72 → Out 9. What is the rule, and what does 64 give?
Each out is its in divided by 8, so the rule is ÷ 8. Then 64 ÷ 8 = 8.
⚠️ Watch out
A single pair rarely fixes the rule. '4 → 28' alone could be × 7 or + 24; only checking '5 → 35' rules out + 24 and confirms × 7.
🌟 Why this matters
Spotting a rule from examples is the doorway to algebra, sequences and coding — every one of them asks you to find the pattern behind the numbers.
Exam tip: Scan for a clean × or ÷; two-step rules are rare at this level.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“In → Out: 48 → 6, 56 → 7, 72 → 9. What is the rule?”
- 1
Compare in with out
Each out is its in divided by 8.
- 2
Check a second pair
56 ÷ 8 = 7 ✓.
- 3
Answer ✓
the rule is ÷ 8
Now you try one — same steps, different story 👇
See it in action
⚡ Time to play!
Two pairs, not one. Then send your number through.
RULE SPOTTER · ROUND 1 OF 3
Step 1 · name the rule
Quick check
✅ VERIFICATION
In → Out: 4 → 28, 5 → 35, 6 → 42. What is the rule?
⚡ Finish the activity above first to answer.
Keep going!