# How one can find a matrix with given condition?

I found the following recreational mathematics problem too hard for me. Can anyone give me hints how to find a solution?

Consider a $4\times 4$ matrix with integer coefficients. Let its elements be $a_{i,j}$ for $1\leq i,j\leq 4$. Now form four sets $A_1$, $A_2$, $A_3$, $A_4$. Say that $A_1$ is the set of elements $a_{i,j}$. Also, $A_2$ is the set of elements $a_{i,j}+a_{i+1,j}+a_{i,j+1}+a_{i+1,j+1}$ i.e. the set of sums of elements of $2\times 2$-subsquares of the matrix. Similarly, denote by $A_3$ the set of sums of $3\times 3$-submatrices and $A_4$ the set containing the sum of all elements of the given $4\times 4$-matrix.

Now, I heard a rumor that one can give an example of 16 integers $a_{i,j}$ such that all integers from 1 to 25 belong to $A_1 \cup A_2 \cup A_3 \cup A_4$. How can one find such an example? It is easy to see that at least some of the elements $a_{i,j}$ must satisfy $1\leq a_{i,j}\leq 25$ but there are still so many elements that finding such a matrix by brute force seems impossible. I was wondering if genetic algorithms or simulated annealing work for such a problem but I don't have enough experience to implement that.

The best I know is that getting all integers from 1 to 24 is possible, for instance using the matrix:

```
-42 22 23 7
13 11 -32 14
-23 16 15 8
19 9 -22 1
```