Example: 108 singers participate in the arts festival, preparing to arrange $m$ performances, with 4 of them performing on stage each time, requiring any two of the 8 singers to perform on the same stage the same number of times. Please design a scheme to minimize the number of performances $m$.
(1996 CMO Problem)
Let the number of times any pair of singers perform together be $r$. Since 8 singers can form $C_{8}^{2}=28$ pairs. If a pair of singers $a_{i}, a_{j}$ perform together in the $k$-th performance, then $\left\{a_{i}, a_{j}, k\right\}$ forms a triplet. Let the number of such triplets be $S$, thus $S=r C_{8}^{2}=28 r$.

On the other hand, each performance involves 4 singers, forming $C_{4}^{2}$ triplets. Over $m$ performances, a total of $m C_{4}^{2}$ triplets can be formed, so $S=m C_{4}^{2}=6 m$. Therefore, $6 m=28 r$, or $3 m=14 r$.
Since $(3,14)=1$, it follows that $14 \mid m$, hence $m \geqslant 14$.
Below is a construction showing that $m=14$ (and thus $r=3$) is achievable (the numbers 1 to 8 represent the 8 singers, and each set of 4 numbers in parentheses represents 4 singers performing together):
$$
\begin{array}{l}
\{1,2,3,4\},\{1,2,5,6\},\{1,2,7,8\},\{1,3,5,7\},\{1,3,6,8\}, \\
\{1,4,5,8\},\{1,4,6,7\},\{2,3,5,8\},\{2,3,6,7\},\{2,4,5,7\}, \\
\{2,4,6,8\},\{3,4,5,6\},\{3,4,7,8\},\{5,6,7,8\} .
\end{array}
$$

In conclusion, the minimum value of $m$ is 14.