We claim that $n$ works iff $n \equiv 0,1\pmod 7$. This gives an answer of $\boxed {26}$ Necessity: Taking the obvious graph interpretation, suppose that the out-degrees of the vertices are $4d_i$ for $1\leq i \leq n$ Note that by the condition the $i$-th vertex has in-degree $(n-1-4d_i)+d_i$. We have sum of out-degree's is the same as the sum of in-degree's. This gives : $$\sum_{i=1}^n 4d_i = \sum_{i=1}^n (n-1-3d_i) \implies 7 \sum_{i=1}^{n}d_i= n(n-1)$$$$ \implies 7 \mid n(n-1)$$ Construction: For $n=7k$ , arrange the kids in a circle and number the kids from $0$ to $7k-1$ in clock-wise sense. The kid numbered $0$ will have out-degree $0$. All the other kids will have out-degree exactly $4k$ and they will send message to their $4k-1$ nearest clockwise neighbours and send a message to $0$. This ensures that every kid (except 0) receives exactly $4k-1$ messages , so for each kid, there are $k$ kids who both send and receive messages from it. For $n=7k+1$ , arrange the kids in a circle and number the kids from $1$ to $7k+1$ in clock-wise sense. All the kids will have out-degree exactly $4k$ and they will send message to their $4k$ nearest clockwise neighbours. This ensures that every kid sends and receives exactly $4k$ messages , so for each kid, there are $k$ kids who both send and receive messages from it. We are done $\blacksquare$

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Ey this is my solution

Youcopyme wrote: Ey this is my solution Ya me too A directed graph interpretation made this problem simple.

Can someone explain further why $i$-th vertex has in-degree $(n-1-4d_i) +d_i$ and $\sum_{i=1}^n 4d_i= \sum_{u=1}^n (n-1-3d_i)$? I have hard time understanding this

Steve12345 wrote: We have a group of $n$ kids. For each pair of kids, at least one has sent a message to the other one. For each kid $A$, among the kids to whom $A$ has sent a message, exactly $25 \% $ have sent a message to $A$. How many possible two-digit values of $n$ are there? Proposed by Bulgaria Should every kids send at least one message? If so, I guess that will make construction different when $n=7k$.