# Revision history [back]

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. 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 $$\sum_{i=1}^{25} p_{i,j} \geq = 1\qquad\text{for }j=1,2,\dots,25.$$ 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  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.