How do you solve truth-teller and liar puzzles?

Assign each person a status, assume one, follow the consequences, and discard the case if you reach a contradiction.

Are knights and knaves puzzles on the TMUA?

The specification does not name them. They are practice for the logic of 'and', 'or', 'not' and multi-step reasoning, which it does list.

How do you negate a statement made by a liar?

A liar's statement is false as a whole. Negate the whole statement, not just part of it.

TMUA Truth-Teller and Liar Puzzles

In a truth-teller puzzle, some people always tell the truth and others always lie. The specification does not name this puzzle type, but it trains the 'and', 'or', 'not' and multi-step reasoning it lists.

Paper 2 topic. Specification codes: Arg1, Prf5. Last checked 12 November 2026.

Method

Give each person a status: truth-teller (T) or liar (L).
A truth-teller's statement is true. A liar's statement is false.
Try one assumption, follow it through, and look for a contradiction.

Handling statements

The negation of 'at least one of us is a liar' is 'none of us is a liar'. Take care with 'and', 'or' and 'exactly one'.

Check at the end

Once you have an answer, test every statement against the assigned statuses. A unique answer must pass all of them.

Worked examples

Example 1

$A$ says, 'At least one of us is a liar.' Each of $A$ and $B$ is a truth-teller or a liar. What are they?

Show the worked solution

If $A$ were a liar, the statement would be false, but then $A$ is a liar and the statement is true. Contradiction. So $A$ is a truth-teller, the statement is true, and $B$ is a liar.

Answer: $A$ truth-teller, $B$ liar

Example 2

$A$ says, ' $B$ is a truth-teller.' $B$ says, 'We are of opposite types.' Find both.

Show the worked solution

If $A$ is a truth-teller then $B$ is, so $B$ 's statement is true, but it would claim they differ. Contradiction. If $A$ is a liar then $B$ is a liar, so both are liars and $B$ 's claim is false, which is consistent.

Answer: Both are liars

Example 3

$A$ : ' $B$ is a liar.' $B$ : ' $C$ is a liar.' $C$ : ' $A$ and $B$ are both liars.' Find all three.

Show the worked solution

Suppose $B$ is a liar. Then $A$ is truthful and $C$ is truthful, but $C$ says $A$ is a liar. Contradiction. So $B$ is truthful, $C$ is a liar, and $A$ (who called $B$ a liar) is a liar.

Answer: $A$ liar, $B$ truth-teller, $C$ liar

Common mistakes

Reading a liar's statement as 'the opposite of what they say about one person' without negating the whole statement.
Stopping at the first consistent case without checking for others.
Mishandling 'and', 'or': the negation of 'both' is 'at least one is not'.
Forgetting to check every person's statement at the end.

Test yourself on this topic with our free practice questions.

Quick answers

Practise it under timed conditions

A free timed sample paper with a worked solution for every question. No sign-up. For two full papers and a weak-spot drill, see Full Access.

Free sample paper

TMUA Truth-Teller and Liar Puzzles

Method

Handling statements

Check at the end

Worked examples

Common mistakes

Quick answers

Practise it under timed conditions

Related topics