A liars and truth-tellers puzzle gives you a few people who each say something, and asks who always tells the truth and who always lies. On Yukti, a daily logic puzzle at playyukti.com, they are knights and knaves on an island, or honest and faulty robots in a workshop. Below: where it comes from, the method and a worked example.
Kind of thinking: Deduction
What is a liars and truth-tellers puzzle?
Everyone on the island is one of two kinds. Knights always tell the truth; knaves always lie. Each islander says one thing, about themselves or the others. A statement on its own doesn't tell you whether it is true: the answer is the one way of marking everyone that makes every statement fit at once.
Gentle days have three islanders, and the week adds more. A statement can say what kind someone is, whether two are the same kind, or how many of a group are knights or knaves; on Tough days, parts are joined with and, either-or and if-then. Robot days are the same puzzle in a workshop: honest robots always tell the truth and faulty ones always lie.
Where does it come from?
The names come from Raymond Smullyan (1919–2017), an American mathematician and logician who was also a concert pianist and a magician. His book What Is the Name of This Book? The Riddle of Dracula and Other Logical Puzzles (1978) takes the reader to an island where every inhabitant is a knight or a knave.
Smullyan's rules are strict: two kinds, no exceptions, and every statement either true or false. That strictness is what lets Yukti's solver check every way of marking the islanders and prove that each puzzle has exactly one answer.
How do you solve one?
Five habits do most of the work:
- Suppose, then follow. Pick one islander and suppose they are a knight. Follow what that makes the others. If you reach a contradiction, they are a knave.
- Read every statement two ways. If the speaker is a knight, the statement is true. If the speaker is a knave, its opposite is true.
- Nobody can say "I am a knave". A knight would be lying and a knave would be telling the truth. So if A says "We are both knaves", A must be a knave, the statement is false, and the other one is a knight.
- "Same kind" points at the other one. If B says "A and I are the same kind", A is a knight whatever B is: a knight's statement is true, so A matches B; a knave's is false, so A differs from B, and that makes A a knight too. "Different kinds" makes A a knave.
- Mark only what the statements force. Start with whoever the most statements are about, and check every statement before you move on.
A worked example: three islanders
A says: "B is a knave." B says: "C and I are the same kind." C says: "A is a knight."
- Start with B. Whatever B is, B's statement makes C a knight (habit 4).
- C is a knight, so what C says is true: A is a knight.
- A is a knight, so what A says is true: B is a knave.
- Check B: a knave's statement must be false. C is a knight and B a knave, so they are not the same kind, and B's statement is false. Everything fits.
Three marks, and none of them a guess. Starting with A or C would have meant supposing; B's statement settles C at once.
Tough days: and, either, if
On Tough days, statements join parts together. Three facts help:
- A knave's "and" has at least one false part. A knave's "either… or" has both parts false.
- "If P, then Q" is false only when P is true and Q is false. So whoever says "If I am a knight, then Q" is a knight, and Q is true. For a knave, "I am a knight" is false, which would make the whole statement true, and knaves can't say true things.
- Counts: try both kinds for the speaker. "Exactly one of us is a knave" from a knight means exactly one; from a knave, any other number.
How does Yukti help when you are stuck?
A mark that doesn't fit is stopped, and the board says which statement it breaks. It still counts as a step. A nudge asks you a question, such as "Suppose A is a knight. What would that make B?". The day after, yesterday's route marks everyone in order, with the reason for each mark.
Sources
Read on 26 October 2026: Knights and Knaves (Wikipedia); Raymond Smullyan (Wikipedia).