3. A traveler visited a village where every person either always tells the truth or always lies. The villagers stood in a circle, and each one told the traveler whether the neighbor to their right was truthful or a liar. Based on these statements, the traveler was able to determine unequivocally what fraction of the village's population consists of truthful people. Determine what this fraction is.
3. Let $x$ be the fraction of truthful residents. Imagine that all truthful residents became liars, and all liars "reformed." Then the traveler would hear the same thing! Indeed, the truthfulness of any resident has changed, but the truthfulness of the neighbor they are talking about has also changed. But the fraction of truthful residents in this circle is $1-x$. Thus, the traveler cannot distinguish a circle with a fraction of truthful residents $x$ from a circle with a fraction of truthful residents $1-x$. Therefore, he could determine the fraction of truthful residents only if $x=1-x$. But this means that $x=1 / 2$.

Comment. Let's number the residents clockwise and set $x_{i}=1$ if the $i$-th resident is a liar, and $x_{i}=0$ otherwise. Then the $i$-th resident will tell the traveler $x_{i}+x_{i+1}$, where the addition is performed modulo 2 (i.e., $0+0=0, 0+1=1, 1+1=0$). Therefore, the information received by the traveler can be understood as a system of linear equations over the field of two elements. See also fact 25.