This is the description of the problem:
There are 3 tribes on an island (tribes A, B, C) that live in separate villages. It is known that tribe A consists only of truth-tellers, the members of tribe B start every conversation with a true sentence and then always lie, and the members of tribe C start every conversation with a true sentence but then alternate between false and true statements.
One day the fire department (conveniently situated in some central location with convenient access to all three villages) received an emergency call from one of the inhabitants of the island:
– There is a fire in one of the villages!
– Which village is on fire?
– Our village is!
– Which is?
– Village C…
At this moment the connection broke. Where should the fire engine go?
My attempt
Consider implications by considering each member of tribe separately:
If there would be a member from tribe "A", then the 2nd response would imply the response to the 3rd question should be "A" since from the description of the problem it is assumed members of the specified tribes are within the region of their tribes and not in other regions. However, the response "village C" was obtained. Hence, it is impossible member from tribe "A" called the phone.
In the case of a member from tribe "C" there are 2 possibilities of "alternating truths and falsehoods", namely: i) the first response after true response is directly responded as true and the alternating sequence is generated, ii) the first response after true is response is directly responded as false and the alternating sequence is generated.
In the case of situation (i), the response to 2nd question would imply the truth is that "not our village is in fire". Thus, either village A or village B is on fire. However, from 3rd response it would imply the response it is valid, hence there would be a contradiction between outcome obtained from 2nd response and outcome from 3rd response.
In the case of situation (ii), from 2nd response it would imply "village C is on fire". From 3rd response there would be a consequence "either village A or village B is on fire". This is a contradiction between outcomes derived.
- In the case of member from tribe "B", 2nd response would imply "the village B is not on fire". It implies either village "A" or village "C" is on fire. From 3rd response there would be an implication either village "A" or village "B" is on fire. By taking all truths to be consistent from all responses, it would imply only the case of village "A" yields consistent logical structure.
Based on the above reasoning, it would therefore imply that only member from tribe "B" must have called. Thus it can be deduced village "A" is on fire. Hence, firefighters should be sent to village "A".
Question
I would appreciate if anyone would comment on my solution or propose alternative solution to that problem.