If $\Diamond4_7=\Diamond1_{8}$ and $\Diamond$ represents a digit, solve for $\Diamond$.
Converting everything to base ten: \begin{align*}
\Diamond4_7&=\Diamond1_{8}\\
\Diamond\cdot7^1+4\cdot7^0&=\Diamond\cdot8^1+1\cdot8^0\\
7\Diamond+4&=8\Diamond+1\\
\Diamond&=\boxed{3}.
\end{align*}