FREE LESSON · Thinking Skills Foundations
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If each rule leads to the next, the end of the chain must be true.
A 'must be true' answer passes two tests. First, you can reach it from a fact, one rule at a time. Second, you can't make it false without breaking a rule.
'Must be true' means true every time the facts are true. Start with a fact you're given, then use one rule at a time. Everyone who handles fragile exhibits (museum objects that break easily) wears gloves. Imani handles fragile exhibits. So Imani must wear gloves.
Rules only go one way. 'Everyone who wears gloves works in the north room' takes you from gloves to the north room. It doesn't take you back. 'You can enter only if you have a pass' means everyone who goes in has a pass. It doesn't mean everyone with a pass goes in. Draw each rule as a one-way arrow.
Some rules say 'no'. Nobody in the north room may use the water station. Now join the rules into a chain. Imani handles fragile exhibits, so she wears gloves. She wears gloves, so she works in the north room. She works in the north room, so she can't use the water station.
To knock out a tempting answer, think of one person who keeps every rule true but makes the answer false. Sam works in the north room but only handles clay. Every rule still works for Sam. So 'everyone in the north room handles fragile exhibits' isn't always true. One example like Sam is enough.
Watch for answers that say more than the facts. The facts don't say Imani is the only one with gloves. They say nothing about Imani and clay. An answer that might be true isn't good enough. Pick the one that is true every time.
Must be true
True every time the facts are true, with no exceptions.'Likely' or 'could be' isn't enough.One-way rule
A rule that takes you from one thing to another, but not back.Gloves lead to the north room. The north room doesn't lead to gloves.Only if
Tells you what's needed. 'Enter only if you have a pass' means everyone who goes in has a pass.Having a pass doesn't mean you go in.Counterexample
One example that keeps every rule true but makes an answer false.One is enough to knock out a 'must be true' answer.'No' rule
A rule that says a group can't do something.Nobody in the north room may use the water station, so Imani can't either.The moveFollow each rule forward, one link at a time. Never run a rule backwards.
show
Watch how a strong thinker proves the answer, one rule at a time.
Read the problem, then follow the steps. You'll see the answer, the trap, and what to do next time.
The problem: three rules, one of them a 'no' rule
Question: Imani handles fragile exhibits. Which statement must be true?Museum volunteers
These rules apply to every volunteer:
Imani handles fragile exhibits.
Which statement must be true?
AImani works in the north room and is not allowed to use the water station.
BEveryone in the north room handles fragile exhibits.
CImani is the only volunteer who wears gloves.
DEveryone who cannot use the water station handles fragile exhibits.
EImani carries clay.
Work out what kind of answer you need.
'Must be true' means true every time the rules are true. So I start with the fact about Imani and follow each rule forwards. Then I test the other answers by looking for someone who breaks them.
Use one rule at a time.
See why a tempting answer is wrong.
Tempting answer: B, 'Everyone in the north room handles fragile exhibits.'
B runs the rules backwards. Think of Sam, who works in the north room and only handles clay. Every rule is still true, but B is false.
C adds 'only'. No rule says Imani is the only one with gloves. E might be true, but no rule says it must be.
Fix: start from Imani and only follow the rules forwards.
Say the answer and the shortest proof.
A. Imani handles fragile exhibits, so she wears gloves. Glove wearers work in the north room. Nobody in the north room may use the water station. So Imani works in the north room and can't use the water station. No rule was used backwards.
What to remember for any 'must be true' question.
A 'must be true' answer works every time the rules are true. Follow the arrows forwards: fragile exhibits → gloves → north room → no water station. To knock out an answer like B, find one example like Sam. A 'no' answer counts just as much as a 'yes' answer. Watch for answers with 'all', 'only' or a rule turned around.
What to do on your next question.
| In this question | Next time | Why it helps |
|---|---|---|
| Started with Imani handling fragile exhibits. | Circle the fact you're given first. | You need a fact to start the chain. |
| Went through gloves and the north room to 'no water station'. | Draw an arrow for each rule, including 'no' rules. | You won't skip a step. |
| Used Sam to knock out answer B. | Try to make a tempting answer false while every rule stays true. | One example is enough. |
compare
A tempting answer often says more than the facts.
Two students answered the same question. One followed every rule. The other didn't.
The whole response
The backwards answer
Everyone with a green wristband signed up for kayaking, so Leo must be the only camper who took a swim test.
The full chain
Leo signs up for kayaking, so he takes a swim test, so he gets a green wristband, so he can't go on the night hike. No rule says he is the only one who took the test.
guide
One small step with help before you go solo.
Look back at the museum problem. Now try one like it.
At a race, every marshal wears an orange vest. Everyone in an orange vest is on the field team. Nobody on the field team may go into the scoring room. Jae is a marshal. What must be true about Jae? Why isn't 'everyone on the field team is a marshal' always true?
Want feedback on your own answer? Get started to practise with instant marking.
At a community garden, anyone carrying seed trays wears a green apron. Every green-apron helper works in the glasshouse. Nobody working in the glasshouse is allowed to use the hose. Ava carries seed trays.
Which statement must be true?
At a school observatory, a pupil may borrow the telescope only if they have a star card. Every star-card holder attended night training. Everyone who attended night training knows the safety code. Oliver borrowed the telescope.
Which statement must be true?
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