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, please!)
Alright, "<span>of the 25 residents", bingo, we have it, 25 wait, 26 people use my base?, wat
ok, so um
anyway, 13 use both, so 13 of the 26 using base 10 are also in the 22 using base 14 so 26+22-13=35 people using bases and then there is 1 resident not using anything 35+1=36
36 residents
wait, we had 'of the 25 residents, 13 use both and 1 can't use either' what happened to the other 11, maybe they are using one or the other?
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 $2B + 13 = 2 \cdot 11 + 13 = \boxed{35}$.
