Create a 3x3 grid of unique integers squared, each row, column, diag. must equal N

chris84567

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Not sure if this is the correct category, if its not sorry.

If you create a 3x3 grid of unique integers squared, each row, column and diagonal has to be equal to N.
What is one solution to this problem and what is a general solution to this problem?

n
a2b2c2n
d2e2f2n
g2h2i2n
nnnn

Using systems of equtions I was able to find that e2= 1/3n.

a2 +b2 +c2 =n
-(a2 +e2 +i2 =n)
-(b2 +e2 +h2 =n)
-(c2 +e2 +g2 =n)
g2 +h2 +i2 =n
=
-3e2 =n

After this I have been able to write everything in terms of a, c and n but don't know where to go next.


n
a2b2c2n
d21n/3f2n
g2h2i2n
nnnn


Thanks for any help you can offer.
 
Last edited:
First thing to notice is that you have 9 unknowns, but only 8 equations. So you won't, in general, have a single unique solution. You'll probably have infinitely-many solutions.

What you did is fine. However, each equation can be used to reduce the number of variables, one at a time.

a^2 + b^2 + c^2 = n

a^2 = n - (b^2 + c^2)

In principle you could just substitute this in everywhere that you see a^2, and that variable will have been eliminated. Repeat.
 
In principle you could just substitute this in everywhere that you see a^2, and that variable will have been eliminated. Repeat.

I have substituted and got every thing in terms of a and c.

Im in school right now but when I get home I'll share what I got.
 
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