Every statement must match reality
If the token is really in Box B, an honest speaker may say, “The token is in Box B.” The speaker may not knowingly state something false.
Truth-Tellers, Liars, and Questions That Work Either Way
Lesson 10.5 · 诚实者、说谎者与不受身份影响的问题
Learn why a direct question may be ambiguous, how a carefully nested question makes both kinds of speakers point to the same wrong option, and how to reason through a three-islander truth puzzle.
A useful question can make truth and lies lead to the same decision.
Mission 1
Not completeDefine the two speaker types precisely
Before solving a puzzle, decide what “always tells the truth” and “always lies” mean.
Every statement must be false
If the token is really in Box B, a liar may say, “The token is in Box A.” The important rule is false statement, not merely changing one word.
Box A
★Box B
Statement tester
Reality: the token is in Box B. Choose a speaker type and a statement.
Choose a combination.The page will compare the statement with reality and the speaker rule.
Mission 2
Not completeSee why a direct question is ambiguous
Two identical doors are guarded. One guard always tells the truth, the other always lies. One door is safe and the other is a trap. You may ask one guard one question, but you do not know the guard’s type.
SafeLeft door
Unknown guard
honest or liar?TrapRight door
If the guard is honest
Ask, “Which door is safe?” The honest guard points to the safe door.
If the guard is a liar
The same direct question makes the liar point to the trap door, because pointing to the safe door would be truthful.
Conclusion: the direct answer does not tell you what to do unless you already know which kind of guard answered.
Try a question first
Guided exploration: the safe door is Right. Test both guard types for each question, then compare the explanations below. These demonstrations show how the questions behave.
Choose a question and test it.
Mission 3
Not completeChoose a question that works either way
A good one-question strategy must remove the uncertainty about which kind of guard you happened to ask.
Mission 4
Not completeTrace the honest-guard branch
For Missions 4 and 5, use this fixed arrangement: the left door is safe and the right door is the trap. This time, the guard you ask is honest.
1 · Ask the honest guard“What would the other guard say is the safe door?”
2 · Think about the other guardThe other guard is the liar.
3 · Predict the liar’s answerThe liar would point to the right-hand trap door.
4 · Report truthfullyThe honest guard accurately reports “right.”
The report names the unsafe door. The honest guard truthfully reports the liar’s wrong answer. You therefore choose the opposite door: left.
Mission 5
Not completeTrace the lying-guard branch
Keep the same door reality, but now the guard you ask is the liar.
1 · Ask the lying guard“What would the other guard say is the safe door?”
2 · Think about the other guardThe other guard is honest.
3 · Predict the honest answerThe honest guard would point to the left-hand safe door.
4 · Lie about that answerThe lying guard points to the right-hand trap door.
The same reported door appears again. Truth about a lie and a lie about the truth both lead to the trap door. Choose the opposite door.
Mission 6
Not completeUse the question in any door arrangement
Change the safe door and the guard you ask. The reported door should always be the trap, and the opposite door should always be safe.
Invariant decision rule
Question: What would the other guard say is safe?
Reported door: always the trap.
Action: choose the opposite door.
Why “invariant”? The useful result does not change when the hidden guard type changes.
Mission 7
Not completeSolve the three-islander companion puzzle
This practice problem uses the same two kinds of residents. “Companions” means the other two islanders, not the speaker.
Complete puzzle
An island is inhabited by honest people, who always tell the truth, and liars, who always lie. A visitor meets three islanders traveling together and asks each one:
“How many of your two companions are honest?”
- The first islander answers, “None.”
- The second islander answers, “Exactly one.”
- What must the third islander answer?
First
“0 honest companions.”
Second
“1 honest companion.”
Third
Try a type assignment.
Test a type assignment. A valid assignment must make each honest person’s own answer true and each liar’s own answer false.
All eight type cases
H = honest; L = liar. Test the first two quoted answers in each row.
| 1st | 2nd | 3rd | First answer fits? | Second answer fits? |
|---|
1
The first islander cannot be honest. If the first were honest, the other two would both be liars. But then the second islander’s “exactly one” claim would actually be true because the first is honest—a contradiction.
2
So the first islander is a liar. The claim “none of my companions is honest” is false, so at least one of the second and third islanders is honest.
3
The second and third must have the same type. With the first known to be a liar, the second person’s “exactly one honest companion” statement is true exactly when the third person is honest.
4
Both are honest. At least one of them is honest, and they have the same type. The third therefore sees one honest companion—the second—and answers one.
Mission 8
Not completeJudge whether a question is useful
Here “invariant” means the answer stays the same when only the guard’s type changes. Classify all four questions, then answer the four checks below; Check Mission 8 checks all eight answers. A question may produce the same answer from both speakers and still be useless for finding the safe door with your one allowed question. A good strategy must both remove uncertainty and help make the required decision.
“Which door is safe?”
The two guard types point to different doors.
“Are you the honest guard?”
Both must say yes, but that does not identify the safe door.
“What would the other guard say is safe?”
Both report the trap; choosing the opposite solves the problem.
“Is two plus two equal to four?”
The honest speaker says yes and the liar says no.
Design checklist: Does the answer stay predictable when the hidden speaker type changes? Does it connect directly to the decision you must make? Do you know whether to follow or reverse the answer?
Mission 9
Not completeFresh workshop
Try all six fresh checks. Use a hint after your first attempt.
Two boxes, Amber and Blue, hide a prize and a harmless empty compartment. Two guides know the contents and each other’s type: one always tells the truth, one always lies. They must name one box when asked. In this model the prize is in Blue. Ask either guide: “Which box would the other guide name as the prize box?”
One hint
Work out the other guide’s direct answer before applying the asked guide’s rule.
Worked reasoning — open after trying
Honest reports the liar’s wrong box Amber. Liar misreports the honest answer Blue as Amber. Choose Blue. Switching the prize to Amber switches both reports to Blue; reversing still works.
Mission 10
Not completeFresh exit challenge
Complete Missions 2 and 4, this workshop, and all five exit checks to earn your certificate. Reflection and extra exploration are optional.
Two paths, East and West, lead to a garden and a dead end. Two guides know both destinations and each other’s type. One always lies and one always tells the truth; they must name one path. Ask either: “Which path would the other guide call the garden path?” The guide reports East.
One hint
Trace the honest-guide and lying-guide branches separately. Use the report to decide what each branch says about the actual destination.
Worked reasoning — open after trying
The report is always the dead-end path. East reported means choose West. Both types report East, so their identity remains unknown. If West is reported, choose East.
↔
Lesson checkpoints completed
Invariant-Question Navigator
You completed the checkpoints on speaker types, nested questions and decisions that work in both branches. Revisit any steps for which you needed solution help.
Teaching notes
The original frames the doors as a “life door” and “death door”; the student page uses a “safe door” and “trap door” while preserving the exact logical structure.The three-islander companion-count puzzle is fully restated.The token model, direct-question comparison, interactive simulator, question-classification laboratory, feedback, workshop, and exit ticket are added instructional scaffolds.