KA-0096Truth-tellers and liars
4 points, difficulty 2 of 3Level 3–4about 90sAna 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.