51. A task, if worked on by person A and B together, can be completed in 8 days; if worked on by person B and C together, it can be completed in 6 days; if worked on by person C and D together, it can be completed in 12 days; then, if worked on by person A and D together, it will take $\qquad$ days to complete.
Answer: 24.
Solution: Let the number of days required for A, B, C, and D to complete the task individually be $a, b, c, d$, respectively, then
$$
\begin{array}{r}
\frac{1}{a}+\frac{1}{b}=\frac{1}{8} \\
\frac{1}{b}+\frac{1}{c}=\frac{1}{6} \\
\frac{1}{c}+\frac{1}{d}=\frac{1}{12}
\end{array}
$$
(1) + (3) - (2), we get $\frac{1}{a}+\frac{1}{d}=\frac{1}{8}+\frac{1}{12}-\frac{1}{6}=\frac{3+2-4}{24}=\frac{1}{24}$,

Therefore, A and D working together will take 24 days to complete the task.