Concept 6 of 6
Machine Rules: What Does the Machine Do?
Today's big idea
Two pairs, never one, because one pair can fit two different rules.
A machine hides one rule; find what turns each input into its output.
How it works
👀 What's going on?
A number goes into a machine and a different one comes out. Nobody tells you what happens inside. Watching a few numbers go through is enough to work it out.
💡 Here's the trick
Look at the machine above. Feed it 3 and 8 comes out. Feed it 5 and 10 comes out. Feed it 7 and 12 comes out. Every output is 5 more than its input, so the rule is add 5. Now you can send any number through, and 9 will come out as 14.
🪄 How to do it
- 1.Write each pair down side by side, as the machine above pairs 3 with 8.
- 2.Ask what was done to the first number to reach the second one.
- 3.Check that same rule works on every pair, then use it on the new number.
✍️ Try it
One machine turns 2 into 6 and 4 into 8. What comes out when 10 goes in? Another turns 3 into 9 and 5 into 15. What comes out when 7 goes in?
The first machine adds 4, because 2 and 4 both grow by 4. So 10 becomes 14. The second machine multiplies by 3, because 3 becomes 9 and 5 becomes 15. So 7 becomes 21.
⚠️ Watch out
You might be thinking one pair is enough to name the rule. It is not. Turning 2 into 6 could be adding 4 or multiplying by 3, and only a second pair tells you which.
🌟 Why this matters
A vending machine, a price list and a conversion chart all take something in and give something back by a fixed rule. Spotting the rule lets you predict the next one.
Exam tip: Always test your rule on a second pair before you trust it.
Watch me solve one
Quanta solves it
“Watch how I think through one of these — then you try.”
THE PROBLEM
“A machine turns 5 into 15 and 7 into 21. What comes out when 9 goes in?”
- 1
Line the pairs up
5 pairs with 15, and 7 pairs with 21.
- 2
Ask what was done
Both are three times bigger, so the rule is multiply by 3.
- 3
Use it on the new number
9 times 3 gives 27.
- 4
Answer ✓
27
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
A machine turns 4 into 12 and 6 into 18. What comes out when 8 goes in?
⚡ Finish the activity above first to answer.
Keep going!