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.
 A: Label the squares $a,b,c$ across the top and so on and call the sum of the subsquares $s$.  If you add the four subsquares you get $4s=a+2b+c+2d+4e+2f+g+2h+i$.  We know the sum of all the numbers is $45$, so we have $4s-45=b+d+f+h+3e$ To make $s$ as small as possible, we clearly want $e=1$ and $b,d,f,h=2,3,4,5$ in some order.  This makes $4s-45=17, s=15.5$  As $s$ must be an integer, the minimum $s$ is $16$.  To make $s$ as large as possible, we want $e=9, b,d,f,h=5,6,7,8$ giving $4s-45=53, s=24.5$  We have shown that the minimum is at least $16$ and the maximum is at most $24$.  To finish the proof, you should show a solution for $s=16$ and $s=24$ because it is possible there is another constraint we have not considered.  It is also still possible there is no solution at all.
A: 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
$$
\begin{array}{ccc}
\left(
\begin{array}{ccc}
 5 & 3 & 8 \\
 7 & 1 & 4 \\
 6 & 2 & 9 \\
\end{array}
\right) &\left(
\begin{array}{ccc}
 3 & 6 & 8 \\
 7 & 1 & 2 \\
 4 & 5 & 9 \\
\end{array}
\right) &\left(
\begin{array}{ccc}
 2 & 6 & 7 \\
 9 & 1 & 4 \\
 3 & 5 & 8 \\
\end{array}
\right)\\[3mm]
\left(
\begin{array}{ccc}
 1 & 7 & 4 \\
 8 & 3 & 5 \\
 6 & 2 & 9 \\
\end{array}
\right) &\left(
\begin{array}{ccc}
 1 & 6 & 7 \\
 8 & 5 & 2 \\
 3 & 4 & 9 \\
\end{array}
\right) &\left(
\begin{array}{ccc}
 1 & 5 & 6 \\
 8 & 7 & 3 \\
 4 & 2 & 9 \\
\end{array}
\right)\\[3mm]
\left(
\begin{array}{ccc}
 2 & 3 & 5 \\
 9 & 8 & 6 \\
 4 & 1 & 7 \\
\end{array}
\right) &\left(
\begin{array}{ccc}
 1 & 5 & 3 \\
 9 & 8 & 7 \\
 4 & 2 & 6 \\
\end{array}
\right) & \left(
\begin{array}{ccc}
 1 & 6 & 2 \\
 8 & 9 & 7 \\
 4 & 3 & 5 \\
\end{array}
\right)
\end{array}
$$
