Start with the person everyone talks about
Each person always tells the truth or always lies. Nobody changes part way.
Pick the person the most statements are about. Assume that person tells the truth. Follow every statement to what it forces. If you reach a contradiction, the assumption was wrong, so that person lies. Then go through it again the other way round.
One thing to watch. A liar’s statement is false as a whole, which is weaker than every part of it being false. If a liar says “A and B both lie”, all you know is that at least one of them tells the truth. You do not know which.
Look for a line about somebody else first. Say A claims “I tell the truth”. If A tells the truth, the line is true, which fits. If A lies, the line is false, which also fits. Nothing is ruled out. Now say A claims “I am a liar”. A truth teller cannot say it, because the line would then be false. A liar cannot say it either, because the line would then be true. Nobody can say it. These two self-referential statements give no usable clue about the board.
This practice never puts one of those on a board. Every statement here is about someone else, so you can start with the person the most statements are about.
1Count how many statements are about each person.
2Assume the most-named person tells the truth.
3Follow each statement. A contradiction means the other way round.
One board, worked both ways
A is named by B and by C, so start with A.
Assume A tells the truth. Then A’s statement is true, so B lies and D lies. Now read B. B says “Exactly one of A and D lies”. On this assumption A tells the truth and D lies, so exactly one of them does lie, so B’s statement is true. A truth teller’s statement is true and a liar’s is false, so B cannot be a liar saying something true. This way round breaks.
So A lies. A’s statement is false. Careful here: “B and D both lie” being false only means at least one of them tells the truth. It does not mean both do.
C says “A lies”, which is true, so C tells the truth. D says “C tells the truth”, which is now true, so D tells the truth. B says “Exactly one of A and D lies”. A lies and D does not, so that is true and B tells the truth.
So A lies, and B, C and D all tell the truth. Every statement fits.
One way round fits and the other breaks. That is the whole method.
Your turn
B is named by A and by D, so B is the quickest place to start.
Assume B tells the truth, follow that to what it forces, and see whether anything breaks.
Try it
Does B tell the truth or lie?
- B tells the truth
- B lies
Show the answer
B lies
Right. Assuming B tells the truth makes D a liar, then C a liar, then A a truth teller. A says B and C both lie, which a truth teller cannot say while B tells the truth. That breaks, so B lies.
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Practice Truth tellers