The nativos in circle

1. Each answer does not directly tell you “who is who,” but it does tell you something about the relationship between two neighbors.
2. A truth-teller and a liar respond the same when their right neighbor is from the same tribe.
3. The answers determine the configuration only up to swapping both tribes completely.

Let's call V a truth-teller and M a liar. When a native answers the question about his right neighbor, this happens:

• if both are from the same tribe, answer yes;
• If they are from different tribes, answer no. That is to say: the answers do not identify who is truthful and who is a liar in an absolute sense; They only indicate whether each pair of consecutive neighbors is of the same type or a different type. Therefore, once a possible distribution has been reconstructed, there is always another equally compatible one: it is enough to exchange all the Vs for M and all the Ms for V. The answers will remain exactly the same. This means that the configuration is only determined up to full complement. But the statement says that, even so, the proportion of liars is fixed in a unique way. The only way for a ratio not to change when exchanging truth-tellers and liars is for both quantities to be equal. Therefore, half of the natives lie and the other half tell the truth.