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Here is a matrix you want:

-21 -20 20 11 
6 2 8 -25 
5 4 9 15 
1 12 -22 16

I found it by first fixing at which squares we get values $\{1,2,\dots,25\}$ (we pick them in $\binom{30}{25}=142506$ ways), and then solving an Integer Linear Program. Here we denote elements of the matrix as $a_{i,j}$, and let $S_i$ be the sums of elements of the $i$-th square ($i=1,\dots,25$). Then we introduce binary indicators $p_{i,j}$ ($i,j=1,\dots,25$) telling whether $S_i = j$ and constrainsts: $$\begin{cases} S_i = \sum_{j=1}^{25} p_{i,j} j,\\ \sum_{j=1}^25 p_{i,j} = 1, \end{cases} \qquad\text{for }i=1,2,\dots,25 $$ and $$\sum_{i=1}^{25} p_{i,j} \geq 1\qquad\text{for }j=1,2,\dots,25.$$ The solution above was solved on $1617$-th selection of squares, within a few minutes.

Here is a matrix you want:

-21 -20 20 11 
6 2 8 -25 
5 4 9 15 
1 12 -22 16

I found it by first fixing selecting at which $25$ squares we get values $\{1,2,\dots,25\}$ (we pick them in $\binom{30}{25}=142506$ ways), and then solving an Integer Linear Program. Here In our ILP, we denote elements of the introduce integer variables $a_{i,j}$ denoting matrix as $a_{i,j}$, elements, and let $S_i$ be the sums of elements of the $i$-th square ($i=1,\dots,25$). Then we introduce binary indicators indicator variables $p_{i,j}$ ($i,j=1,\dots,25$) telling whether $S_i = j$ j$, and constrainsts: constraints: $$\begin{cases} S_i = \sum_{j=1}^{25} p_{i,j} j,\\ \sum_{j=1}^25 \sum_{j=1}^{25} p_{i,j} = 1, \end{cases} \qquad\text{for }i=1,2,\dots,25 $$ and $$\sum_{i=1}^{25} p_{i,j} \geq 1\qquad\text{for }j=1,2,\dots,25.$$ The solution above was solved obtained on $1617$-th selection of squares, within just a few minutes.minutes of running my Sage code.

Here is a matrix you want:

-21 -20 20 11 
6 2 8 -25 
5 4 9 15 
1 12 -22 16

I found it by first selecting at which $25$ squares we get values $\{1,2,\dots,25\}$ (we pick them in there are $\binom{30}{25}=142506$ ways), ways to select such 25 squares), and then solving an Integer Linear Program. Programming problem.

In our ILP, we introduce integer variables $a_{i,j}$ denoting the matrix elements, and let $S_i$ be the sums of those elements of in the $i$-th selected square ($i=1,\dots,25$). Then Also, we introduce have binary indicator variables $p_{i,j}$ ($i,j=1,\dots,25$) telling whether $S_i = j$, and constraints: $$\begin{cases} S_i = \sum_{j=1}^{25} p_{i,j} j,\\ \sum_{j=1}^{25} p_{i,j} = 1, \end{cases} \qquad\text{for }i=1,2,\dots,25 $$ and $$\sum_{i=1}^{25} p_{i,j} \geq 1\qquad\text{for }j=1,2,\dots,25.$$ The solution above was obtained on $1617$-th selection of squares, within just a few minutes of running my Sage code.

Here is a matrix you want:

-21 -20 20 11 
6 2 8 -25 
5 4 9 15 
1 12 -22 16

I found it by first selecting at which $25$ squares we get values $\{1,2,\dots,25\}$ there are $\binom{30}{25}=142506$ ways to select such 25 squares), and then solving an Integer Linear Programming problem.

In our ILP, we introduce integer variables $a_{i,j}$ denoting the matrix elements, and let $S_i$ be the sums of those elements in the $i$-th selected square ($i=1,\dots,25$). Also, Essentially we have want $(S_1,S_2,\dots,S_{25})$ be a permutation of $\{1,2,\dots,25\}$. For this purpose, we introduce binary indicator variables $p_{i,j}$ ($i,j=1,\dots,25$) telling whether $S_i = j$, and constraints: $$\begin{cases} S_i = \sum_{j=1}^{25} p_{i,j} j,\\ \sum_{j=1}^{25} p_{i,j} = 1, \end{cases} \qquad\text{for }i=1,2,\dots,25 $$ and $$\sum_{i=1}^{25} p_{i,j} \geq 1\qquad\text{for }j=1,2,\dots,25.$$ The solution above was obtained on the $1617$-th selection of squares, within just a few minutes of running my Sage code.

Here is a matrix you want:

-21 -20 20 11 
6 2 8 -25 
5 4 9 15 
1 12 -22 16

I found it by first selecting at which $25$ squares we get values $\{1,2,\dots,25\}$ there (there are $\binom{30}{25}=142506$ ways to select make such 25 squares), a selection), and then by solving an Integer Linear Programming problem.

In our ILP, we introduce integer variables $a_{i,j}$ denoting the matrix elements, and let $S_i$ be the sums of those elements in the $i$-th selected square ($i=1,\dots,25$). Essentially we want $(S_1,S_2,\dots,S_{25})$ be a permutation of $\{1,2,\dots,25\}$. For this purpose, we introduce binary indicator variables $p_{i,j}$ ($i,j=1,\dots,25$) telling whether $S_i = j$, and constraints: $$\begin{cases} S_i = \sum_{j=1}^{25} p_{i,j} j,\\ \sum_{j=1}^{25} p_{i,j} = 1, \end{cases} \qquad\text{for }i=1,2,\dots,25 $$ and $$\sum_{i=1}^{25} p_{i,j} \geq 1\qquad\text{for }j=1,2,\dots,25.$$ The solution above was obtained on the $1617$-th selection of squares, within just a few minutes of running my Sage code.

Here is a matrix you want:

-21 -20 20 11 
6 2 8 -25 
5 4 9 15 
1 12 -22 16

I found it by first selecting at which $25$ squares we get values $\{1,2,\dots,25\}$ (there are $\binom{30}{25}=142506$ ways to make such a selection), and then by solving an Integer Linear Programming problem.

In our ILP, we introduce integer variables $a_{i,j}$ $a_{u,v}$ ($u,v=1,2,3,4$) denoting the matrix elements, and let $S_i$ be the sums of those elements in the $i$-th selected square ($i=1,\dots,25$). Essentially we want $(S_1,S_2,\dots,S_{25})$ be a permutation of $\{1,2,\dots,25\}$. For this purpose, we introduce binary indicator variables $p_{i,j}$ ($i,j=1,\dots,25$) telling whether $S_i = j$, and constraints: $$\begin{cases} S_i = \sum_{j=1}^{25} p_{i,j} j,\\ \sum_{j=1}^{25} p_{i,j} = 1, \end{cases} \qquad\text{for }i=1,2,\dots,25 $$ and $$\sum_{i=1}^{25} p_{i,j} \geq 1\qquad\text{for }j=1,2,\dots,25.$$ The solution above was obtained on the $1617$-th selection of squares, within just a few minutes of running my Sage code.

Here is a matrix you want:

-21 -20 20 11 
6 2 8 -25 
5 4 9 15 
1 12 -22 16

I found it by first selecting at which $25$ squares we get values $\{1,2,\dots,25\}$ (there are $\binom{30}{25}=142506$ ways to make such a selection), and then by solving an Integer Linear Programming problem.

In our ILP, we introduce integer variables $a_{u,v}$ ($u,v=1,2,3,4$) denoting the matrix elements, and let $S_i$ be the sums of those elements in the $i$-th selected square ($i=1,\dots,25$). Essentially we want $(S_1,S_2,\dots,S_{25})$ be a permutation of $\{1,2,\dots,25\}$. For this purpose, we introduce binary indicator variables $p_{i,j}$ ($i,j=1,\dots,25$) telling whether $S_i = j$, and constraints: $$\begin{cases} S_i = \sum_{j=1}^{25} p_{i,j} j,\\ \sum_{j=1}^{25} p_{i,j} = 1, \end{cases} \qquad\text{for }i=1,2,\dots,25 $$ and $$\sum_{i=1}^{25} p_{i,j} \geq 1\qquad\text{for }j=1,2,\dots,25.$$ The solution above was obtained on the $1617$-th selection of squares, within just a few minutes of running my Sage code.

9 3 6 5 
8 1 4 10 
11 2 16 -23 
19 -15 12 13

Here is a matrix you want:

-21 -20 20 11 
6 2 8 -25 
5 4 9 15 
1 12 -22 16

I found it by first selecting at which $25$ squares we get values $\{1,2,\dots,25\}$ (there are $\binom{30}{25}=142506$ ways to make such a selection), and then by solving an Integer Linear Programming problem.

In our ILP, we introduce integer variables $a_{u,v}$ ($u,v=1,2,3,4$) denoting the matrix elements, and let $S_i$ be the sums of those elements in the $i$-th selected square ($i=1,\dots,25$). Essentially we want $(S_1,S_2,\dots,S_{25})$ be a permutation of $\{1,2,\dots,25\}$. For this purpose, we introduce binary indicator variables $p_{i,j}$ ($i,j=1,\dots,25$) telling whether $S_i = j$, and constraints: $$\begin{cases} S_i = \sum_{j=1}^{25} p_{i,j} j,\\ \sum_{j=1}^{25} p_{i,j} = 1, \end{cases} \qquad\text{for }i=1,2,\dots,25 $$ and $$\sum_{i=1}^{25} p_{i,j} \geq = 1\qquad\text{for }j=1,2,\dots,25.$$ The solution above was obtained on the $1617$-th selection of squares, within just a few minutes of running my Sage code.

PS. Another solution:

9 3 6 5 
8 1 4 10 
11 2 16 -23 
19 -15 12 13

Here is a matrix you want:

-21 -20 20 11 
6 2 8 -25 
5 4 9 15 
1 12 -22 16

I found it by first selecting at which $25$ squares we get values $\{1,2,\dots,25\}$ (there are $\binom{30}{25}=142506$ ways to make such a selection), and then by solving an Integer Linear Programming problem.

In our ILP, we introduce integer variables $a_{u,v}$ ($u,v=1,2,3,4$) denoting the matrix elements, and let $S_i$ be the sums of those elements in the $i$-th selected square ($i=1,\dots,25$). Essentially we want $(S_1,S_2,\dots,S_{25})$ be a permutation of $\{1,2,\dots,25\}$. For this purpose, we introduce binary indicator variables $p_{i,j}$ ($i,j=1,\dots,25$) telling whether $S_i = j$, and constraints: $$\begin{cases} S_i = \sum_{j=1}^{25} p_{i,j} j,\\ \sum_{j=1}^{25} p_{i,j} = 1, \end{cases} \qquad\text{for }i=1,2,\dots,25 $$ and $$\sum_{i=1}^{25} p_{i,j} = 1\qquad\text{for }j=1,2,\dots,25.$$ The solution above was obtained on the $1617$-th selection of squares, within just a few minutes of running my Sage code.

PS. Another solution:

9 3 6 5 
8 1 4 10 
11 2 16 -23 
19 -15 12 13

0 6 21 4 
9 7 -24 17 
14 -25 11 8 
16 15 1 3