In the small hamlet of abaze, two base systems are in common use. also, everyone speaks the truth, and no one changes which base they are using in mid-sentence. one resident said, "26 people use my base, base 10, and only 22 people speak base 14." another said, "of the 25 residents, 13 use both bases and 1 can't use either base." how many residents are there? (use base 10, ! )

Suppose that the first resident speaks in base $B$. Converting to base $10$, his statement implies that $2B + 6$ people uses base $B$ (notice that base $10$ is actually base $1 \cdot B + 0 = B$), and that $2B+2$ people speak using base $B + 4$. Clearly, the second resident is not talking in base $B$, so he must speak base $B + 4$. Then we can also write all of his statements in terms of the variable $B$. It follows that there are $2(B+4) + 5 = 2B + 13$ total residents, of which $(B+4)+3 = B+7$ are bilingual and $1$ cannot use either base. By the Principle of Inclusion-Exclusion, there are $$(2B + 6) + (2B+2) + 1 - (B+7) = 3B + 2$$total people, and setting this equal to $2B + 13$, it follows that $3B + 2 = 2B + 13 \Longrightarrow B = 11$. Thus, the answer is $2B + 13 = 2 \cdot 11 + 13 = \boxed{35}$.