A number $N$ has three digits when expressed in base $7$. When $N$ is expressed in base $9$ the digits are reversed. Then the middle digit is:
$\text{(A) } 0\quad \text{(B) } 1\quad \text{(C) } 3\quad \text{(D) } 4\quad \text{(E) } 5$
Call the number $\overline{abc}$ in base 7.
Then, $49a+7b+c=81c+9b+a$. (Breaking down the number in base-form).
After combining like terms and moving the variables around,
$48a=2b+80c$,$b=40c-24a=8(5c-2a)$. This shows that $b$ is a multiple of 8 (we only have to find the middle digit under one of the bases). Thus, $b=0$ (since 8>6, the largest digit in base 7).
Select $\fbox{A}$ as our answer.
~hastapasta