1 | initial version |

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.

2 | No.2 Revision |

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.

3 | No.3 Revision |

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.

4 | No.4 Revision |

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.

5 | No.5 Revision |

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.

6 | No.6 Revision |

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.

7 | No.7 Revision |

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
```

8 | No.8 Revision |

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
```

9 | No.9 Revision |

Here is a matrix you want:

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

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
```

PS2. There even exists a matrix delivering sums $\{0,1,2,\dots,25\}$, for example:

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

However, the sums $\{1,2,3,\dots,26\}$ are not possible to obtain.

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