![]() | 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,…,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
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
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.