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The cadena of mentiras (7 in circle)

A wheel of affirmations may seem like a verbal maze, but here it all depends on how one phrase compels the next. The pleasure of the problem is in seeing the pattern appear.

There are 7 people sitting in a circle. Each one says exactly: > “My neighbor on the right is a liar.” It is known that each person belongs to one of these two types:

truthful, if he always tells the truth;
liar, if he always lies. Is such a distribution possible?

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Hints

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Think about what a true sentence requires about the neighbor on the right, and what a false sentence requires.
If one tells the truth, his right neighbor lies; If one lies, his right neighbor tells the truth.
The pattern necessarily alternates, and with 7 people the odd cycle makes it impossible.

Solution

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Answer: No, it is impossible. If a person tells the truth when stating “my right neighbor is a liar,” then his neighbor is a liar.
If a person lies in stating that, then his right neighbor is truthful. In both cases, the right neighbor type is the opposite. Therefore, around the circle you must alternate:

$$ V,M,V,M,\dots $$ This alternation can only be closed without contradiction when the number of people is even. With 7 (odd), returning to the beginning requires that the first person be both equal and different from themselves. **Conclusion:** Impossible configuration for 7. $$

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