KA-0017Truth-tellers and liars
5 points, difficulty 3 of 3Level 3–4about 120sAna says "Bo is lying." Bo says "Cal is lying." Cal says "Ana and Bo are both lying." How many of the three are telling the truth?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Assume Ana is telling the truth, then follow it.
2The strategy
Take one speaker, assume they are truthful, and push that through all three statements. If it contradicts itself the assumption was wrong, and that is itself information.
3The full solution
Suppose Ana is truthful. Then Bo lies, so Cal is truthful. But Cal says Ana lies, contradicting the assumption. So Ana lies, Bo is truthful, and Cal lies - which is consistent. Exactly one tells the truth.
Solution
The reliable way
Try Ana truthful. Then Bo lies, so Cal's claim is true, so Cal is truthful. But Cal claims Ana lies, which contradicts the assumption. So Ana lies. Then Bo tells the truth, so Cal lies. Check Cal: Cal claims Ana and Bo both lie, but Bo is truthful, so Cal's claim is indeed false. Exactly one child, Bo, tells the truth. The transferable idea: assume, follow all the way, and let the contradiction do the work.
The elegant way
The three claims form a chain in which truth must alternate, and Cal's double claim leaves only one arrangement standing.
Why this is on the test: Liar puzzles reward pushing one assumption to the end rather than stopping at the first thing that looks consistent.