Magic Table Problem

[JoshDontKnow]

I've encountered a problem that seems very simple, but turned out to just be frustrating
.... and now I seek math experts to help me.



The idea is to enter a number in each of the red and green blocks.

Red blocks must contain negative numbers " < 0 " (Smaller than 0)
Green blocks must contain positive numbers " > 0 " (Greater than 0)

Just ignore all the white (empty) blocks

All the numbers above the light blue line (that contains zeros) must be opposites of the numbers below the line....

The sum of each column must be 0
The sum of each row must be 0

Any solution would be much appreciated ...
but I would like to find the smallest whole number solution ...
furthermore ... if it's not too much trouble:
Could some math expert out there show me how to find a solution to any other grid
that works in the same way...
with different red/green box configurations ... and of any other grid size.

Thank you.

 

[JoshDontKnow]

The Table

Here's the table spreadsheet

 

[Ishuda]

...
All the numbers above the light blue line (that contains zeros) must be opposites of the numbers below the line....
...

If this means what I think it means, i.e. for example the number just above the diagonal in a column must be the negative of the the number just below the diagonal, then there would need to be some sort of (anti)symmetry in the red and green blocks which you do not have.

 

[JoshDontKnow]

Little Solutions

What I mean by opposites is: For example Cell "B:H" is the direct opposite (...or negative) of Cell "H:B"

Here are solutions that I have found.....

but only for a 4x4 table

 

[Ishuda]

What I mean by opposites is: For example Cell "B:H" is the direct opposite (...or negative) of Cell "H:B" ...

O.K. What you mean then is that the matrix

must be symmetric. Not all configurations of the red and green blocks have a solution so are you guaranteed that the configuration of red and green blocks have a solution for what you want?

As a very simple example consider
0RG
R0W
GW0
Where R stands for a red block, G for a green block, W for a white block, and 0 is zero.

 

[JoshDontKnow]

The example you show doesn't have a solution because there is a column with a red block but no green block...
which means the sum of the column cannot be zero, because the column can't contain any positive numbers
to cancel out the negative numbers.....

I think that the table I posted must have a solution because every column contains red and green blocks

This example has a solution. It is symmetric but the one half is the exact negative of the other half:

0RG
G0R
RG0

 

[Ishuda]

You had asked "Could some math expert out there show me how to find a solution to any other grid
that works in the same way" and, although I don't claim to be an expert, what I was suggesting is that there may not be a solution for all cases. Also, to obtain the minimum numbers you at least must have that there be no common divisor of all the numbers as there is in your example above for the 4X4 table.

At the present time, I certainly don't have a method for determining a solution other than a systematic trial & error.

 

[JoshDontKnow]

OK.... I gave the problem to a friend of mine and he actually found a solution.

But the only way he could find the solution was to tweak the table....
thereby I mean he added red and green blocks in the grid to make it balance out.
So technically it isn't a solution to the table I posted....... It's a solution to another configuration.

Here's what he came up with:

Thanks for all the responses though.....
Really appreciate it.

 

[JoshDontKnow]

Solutions

OK .... so a lot of time has been spent on this .......
but I've finally found some solutions.

If anyone out there can find a solution with lower numbers.
Please be so kind as to share.
Thank you.