5.70 It is known that 12 theater troupes are participating in a 7-day drama festival, and it is required that each troupe can see the performances of all other troupes (actors who do not have a performance on that day watch the performances from the audience). How many performances are needed at a minimum?

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［Solution］Obviously, each troupe performs at least 1 show. If Troupe $A$ performs only 1 show in 7 days, then on the day $A$ troupe performs, no other troupes perform, otherwise, $A$ troupe's performance will not be seen. Thus, only one show is performed on that day.

If 3 troupes each perform only 1 show during the festival, then these 3 troupes will occupy 3 out of the 7 days. Therefore, the remaining 9 troupes can only perform in the remaining 4 days. Since these 9 troupes also need to observe each other, the sets of dates on which they perform must not contain each other. However, the number of non-overlapping subsets that can be formed from 4 different natural numbers is at most 6, which is insufficient to accommodate 9 troupes, leading to a contradiction. Therefore, at most two troupes can each perform 1 show, and the rest of the troupes must each perform at least 2 shows, totaling at least 22 shows.
On the other hand, if the performance schedules of the 12 troupes are arranged as follows:
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline Troupe & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline Performance & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 3 & 3 & 4 & 6 & 7 \\
Date & 2 & 3 & 4 & 5 & 3 & 4 & 5 & 4 & 5 & 5 & & \\
\hline
\end{tabular}

This arrangement meets the requirements of the problem, and the 12 troupes perform a total of 22 shows. In summary, the minimum number of shows is 22.