Ava: “Exactly one of us three is a truth-teller.”
Ben: “The lodge is not on the left trail.”
Cal: “Ava and Ben are the same kind: both truth-tellers or both liars.”
Which trail leads to the lodge, and who is telling the truth?
💡 Hint
Begin with the speaker who talks only about the other speakers, not about the paths. Test both possibilities for that statement, and check Ava’s claim against each.
Reveal answer
Take the left trail. Ava tells the truth; Ben and Cal are liars.
Step-by-step solution
- Suppose Cal tells the truth, so Ava and Ben are the same kind. If both are truth-tellers, there are at least two truth-tellers, so Ava’s statement is false. But Ava is a truth-teller. Contradiction.
- If Ava and Ben are both liars while Cal tells the truth, then Cal is the only truth-teller, so Ava’s statement (“exactly one”) is true. But Ava is a liar. Contradiction. So Cal is a liar.
- Since Cal lies, Ava and Ben are of different kinds. Exactly one of them tells the truth, and Cal lies, so there is exactly one truth-teller in total.
- That makes Ava’s statement true, so Ava is the truth-teller and Ben is a liar.
- Ben lied when he said the lodge is not on the left. So the lodge is on the left trail. Checking all 24 combinations of types and trails confirms this is the only consistent one.
Why it fools people
Our attention goes straight to the statement that mentions the trails, because that is the question we were asked. But that statement is the weakest one: it is worthless until you know whether Ben lies. The solution runs in the opposite direction, from the statements about people to the statement about the trails. Ava’s claim, which sounds like a trick, turns out to be the anchor.
Did you get it?
About This Puzzle
Deep in a national forest, three trails leave a single clearing: left, middle and right. Only one of them reaches the lodge before dark. Three local guides are resting at the trailhead, and in this corner of the woods every guide is one of two kinds. Some always tell the truth, and some always lie. You don’t know who is which. Each guide makes exactly one statement, and you have to choose your trail from those three statements alone. There is no clever question to ask here and no way to test anyone. Everything you need is in what they say, and exactly one trail is consistent with it.
How to Think About It
In truth-teller and liar puzzles, every statement does double duty. It tells you something about the world, and it tells you about the speaker, because the statement is true exactly when the speaker is a truth-teller. Start with a statement that talks only about the speakers, not about the trails. Cal’s statement is a good place to begin. Split into cases: suppose it is true, then suppose it is false. In each case, follow the consequences to the end and see whether Ava’s statement agrees with what you’ve assumed. Throw out any case that contradicts itself.
Common Traps
The most common mistake is to start with Ben, because his statement is the only one about the trails, and that is the part you care about. But Ben’s statement is useless until you know whether he lies. Another trap is to treat Ava’s statement as obviously false because it sounds suspicious. Self-referential claims can be perfectly true. Finally, many solvers find one consistent arrangement and stop. You need to check every case, because a well-built puzzle has exactly one survivor, and you should confirm that.
Where This Puzzle Comes From
Puzzles about truth-tellers and liars are an old tradition in logic and became widely known through the books of the American logician and puzzle writer Raymond Smullyan, who called his characters knights and knaves. The famous two-guards puzzle belongs to the same family. This puzzle is an original construction in that tradition, built so that each statement is needed to reach the answer.
Try This Variation
Keep Ava’s and Ben’s statements, but Cal now says: “Ava is a liar.” Can you still be sure which trail leads to the lodge? List every combination of truth-tellers and liars that survives.
Show the answer to the variation
No. Three scenarios survive. In the first, Ava tells the truth, Ben and Cal lie, and the lodge is on the left. In the other two, Ava lies, Ben and Cal tell the truth, and the lodge is on the middle or the right trail. The original Cal statement is what pins the answer down.