Problem
Truth-tellers and liars
Two tribes live on an island: one always tells the truth, the other always lies. Which is which is not stated. An islander is asked his tribe and gives an answer. Work out which tribe is the truthful one.
Reasoning · Grade 6-1 Truth and Falsehood
Your answer
Watch out Believing the answer is tempting, but nobody can say he is from the lying tribe.
How to solve
Strategy Eliminate Possibilities — The question asks which tribe is truthful, so I test the two answers directly: suppose the Deer tribe is the truthful one, and suppose the Fox tribe is the truthful one. Each supposition still leaves the speaker's own tribe open, so each splits into two sub-cases, giving four situations in all -- a small enough list to write out completely. In each situation I check one thing only: does the speaker's honesty match whether his sentence is actually true? A situation where an honest speaker says something false, or a liar says something true, is impossible and gets crossed off. That eliminates one whole supposition on both of its branches, which is a real contradiction and not just an absence of evidence, and it leaves the other supposition standing with a working example to show it is genuinely possible.
1STEP 1
Sort out what can vary, and list every situation
Listing what can vary gives four situations.
2 × 2 = 4 situations
Sorting the islanders into two boxes -- Fox or Deer -- and separately labelling one box 'always truthful' is exactly the classify-into-categories thinking children do with shape sorters, and it turns a riddle into a short list.
K.MD.B.3Make A Systematic List2STEP 2
State the one test that every situation must pass
Test each: could that answer be given?
It is one yes-or-no check applied to two labels on the same person, so a young solver can run it in a single sentence instead of holding a chain of reasoning in mind.
K.MD.B.3Make A Systematic ListEvery situation has to pass one test, and a situation that fails it is discarded without further thought.
Why?
A single sentence the situation makes both true and false is enough to kill it, however consistent the rest looked.
🧱One counterexample refutes an always-claimA claim about every case dies the moment you exhibit one case where it fails, and exhibiting is easier than arguing.
Why?
Once every competing situation has been discarded for a definite reason, the one still standing has to be the truth.
🧱Rule out the rest and the last one is forcedIf every possibility but one has been ruled out, the survivor must be the answer — no further checking is needed.
3STEP 3
Suppose the Deer tribe is the truthful one -- kill the first branch
If Deer are truthful, one branch is a contradiction.
The contradiction is at the level of a name tag: the same person ends up wearing two different tribe tags, which even a kindergartener can see cannot happen.
K.MD.B.3Eliminate Possibilities4STEP 4
Suppose the Deer tribe is the truthful one -- kill the other branch too
The other branch is also a contradiction.
Both branches fail for the same reason, so a young solver only has to spot the pattern once: whoever the speaker is, calling himself a Fox clashes with Deer being the honest tribe.
K.MD.B.3Eliminate Possibilities5STEP 5
Suppose the Fox tribe is the truthful one -- show it actually works
If Fox are truthful, it works out.
Building one complete example and checking every rule against it is what convinces you the surviving case is real, rather than just the case you happened not to break.
K.MD.B.3Eliminate Possibilities6STEP 6
Read off the answer, and notice why the question was fair
So the truthful tribe is the Fox tribe.
Shifting from 'who is this person' to 'what can anyone possibly say' collapses two unknowns into one and shows the answer was forced before the scholar even chose whom to ask.
K.MD.B.3Change Focus Count The ComplementAnswer
the Fox tribe
2 × 2 = 4
The answer names one of the two tribes, which is what the question asked for, and it is not a number, so there are no units or magnitudes to check. The right sanity check is that the reasoning uses every given: both tribes appear, the all-truth / all-lies rule is used in every case, and the speaker's actual sentence is what breaks the tie -- had he said 'I am from the Deer tribe', the same four-situation test would have killed the two Fox-truthful situations instead and left the two Deer-truthful ones standing, so the answer would have flipped to the Deer tribe. Also, the conclusion is stable: it comes out the same whether the speaker was honest or lying, which it must, since the scholar had no way to choose an honest islander on purpose.
Takeaway
Try each answer as a guess and see if it makes someone honest tell a lie -- and remember that nobody can ever admit to being a liar, so the tribe an islander names is always the honest one.
- Sort out what can vary, and list every situation
- State the one test that every situation must pass
- Suppose the Deer tribe is the truthful one -- kill the first branch
- Suppose the Deer tribe is the truthful one -- kill the other branch too
- Suppose the Fox tribe is the truthful one -- show it actually works
- Read off the answer, and notice why the question was fair