<s>[INST] Square 3x3 filled with numbers with every 2x2 subsquare having equal sum. You are given 3x3 square filled with numbers from 1 to 9 without repetition. Every subsquare 2x2 has equal sum. What are the possible sums of those subsquares? I think that all values between 16 and 24 are possible, but I don't know how to prove it. I can check all the possibilites, but I am looking for an elegant and as simple as possible proof.
 [/INST] As a modest complement to Ross Millikan's answer,
I have tested the $9!$ matrices and all the sums from $16$ to $24$ appear,
with the following frequencies:
$$
\begin{array}{c|ccccccccc}
 s & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 \\\hline
\sharp & 16 & 40 & 40 & 64 & 56 & 64 & 40 & 40 & 16 \\
\end{array}
$$
And here are examples for each sum
$$
\begin{array}{ccc}
\left(
\begin{array}{ccc}
 5 & 3 & 8 \\
 7 & 1 & 4 \\
 6 & 2 & 9 \\
\end{array}
\right) &\left(
\begin{array}{ccc}
 3 & 6 & 8 \\
 7 & 1 & 2 \\
 4 & 5 & 9 \\
\end{array}
\right) &\left(
\begin{array}{ccc}
 2 & 6 & 7 \\
 9 & 1 & 4 \\
 3 & 5 & 8 \\
\end{array}
\right)\\[3mm]
\left(
\begin{array}{ccc}
 1 & 7 & 4 \\
 8 & 3 & 5 \\
 6 & 2 & 9 \\
\end{array}
\right) &\left(
\begin{array}{ccc}
 1 & 6 & 7 \\
 8 & 5 & 2 \\
 3 & 4 & 9 \\
\end{array}
\right) &\left(
\begin{array}{ccc}
 1 & 5 & 6 \\
 8 & 7 & 3 \\
 4 & 2 & 9 \\
\end{array}
\right)\\[3mm]
\left(
\begin{array}{ccc}
 2 & 3 & 5 \\
 9 & 8 & 6 \\
 4 & 1 & 7 \\
\end{array}
\right) &\left(
\begin{array}{ccc}
 1 & 5 & 3 \\
 9 & 8 & 7 \\
 4 & 2 & 6 \\
\end{array}
\right) & \left(
\begin{array}{ccc}
 1 & 6 & 2 \\
 8 & 9 & 7 \\
 4 & 3 & 5 \\
\end{array}
\right)
\end{array}
$$
</s>