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I was thinking of the following recreational math problem:

We have a $4\times 4$ square filled with integers $a_{1,1},...,a_{4,4}$. It has $30$ sub-squares $A_{i,j,k}$, corners of the form $a_{i,j},a_{i,j+k},a_{i+k,j}, a_{i+k,j+k}$, such that sum of the elements of $A_{i,j,k}$ are denoted by $s_1,\cdots ,s_{30}$.

How can I show that it is possible to choose the integers $a_{i,j}$ such that

$\{1,2,\cdots,24\}\subset \{s_1,\cdots,s_{30}\}$?

And is it possible to prove that always $\{1,2,\cdots,25\}\not\subset \{s_1,\cdots,s_{30}\}$?

And what kind of methods there are to find those numbers $a_{i,j}$ or to prove that their absolute value can't be too large?

My thoughts are that one can form a system of linear equations to solve the problem but there are over $24!$ cases to check whether the system has a solution.

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So your sub-squares include k=1 case $A_{i,j,1}=[a_{i,j}]$ – Nate Iverson Jun 4 '12 at 14:42
Yes, that is right. – student Jun 4 '12 at 14:47

To show that 24 integers are possible is solved in

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