Some speakers always tell the truth and some always lie. The method is to assume one speaker is truthful, follow the consequences all the way, and see whether anything contradicts. If it does, that assumption was wrong, which is itself information.
You'll recognise it when…
“knights always tell the truth and knaves always lie”
●●○4 points, difficulty 2 of 34 ptLevel 3–4about 90s
Ana says: "Bo and I are both liars." Each child is either always truthful or always a liar. Who is telling the truth?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Could a truthful person say that sentence?
2The strategy
Suppose Ana is truthful and read her own sentence back. If it makes her a liar, the supposition is impossible - and knowing she lies tells you her sentence is false.
3The full solution
If Ana were truthful, her sentence would make her a liar - impossible. So Ana lies, and her sentence is false: they are not both liars. Ana is one, so Bo is not.
Solution
The reliable way
Suppose Ana is truthful. Then her sentence is true, so she is a liar - a contradiction. So Ana is a liar, and her sentence is false. Both of us are liars being false means they are not both liars. Ana is one, so Bo is not: Bo tells the truth. Check: Bo being truthful makes Ana's claim genuinely false, which is exactly what a liar should say. The transferable idea: a statement that makes its own speaker a liar settles who is lying in one step.
The elegant way
Nobody truthful can ever call themselves a liar, so Ana lies - and the negation of her sentence then hands you Bo.
Why this is on the test: A self-referring statement is the fastest kind to resolve, and students who hunt for extra clues rather than reading it back miss that.
Where it goes wrong
Each of these is written the way a student would say it to themselves.
“I assumed a speaker was truthful and stopped as soon as things looked consistent, without checking the rest.”
“I forgot that a liar's statement being false tells me something definite.”
“I tried to reason about all the speakers at once instead of assuming one and following it through.”
Every problem in this skill
●○○3 points, difficulty 1 of 33 ptOne step from a skill you know.
Nothing at this tier yet for LOG-02.
●●○4 points, difficulty 2 of 34 ptTwo steps, or one step plus a trap.