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### Re^4: magic squares

by Limbic~Region (Chancellor)
 on Apr 08, 2009 at 15:28 UTC ( #756364=note: print w/replies, xml ) Need Help??

in reply to Re^3: magic squares
in thread magic squares

tilly,
Yes, this is exactly what I was guessing at. There are also constraints as to what x, y and z can be within the confines of this puzzle. I didn't have a chance last night to work on this though because the following entered my mind as I was leaving work
```X+Y      X+Z     X-Y-Z
X-2Y-Z   X       X+2Y+Z
X+Y+Z    X-Z     X-Y

Terms: X, X+Y, X-Y, X+Z, X-Z, X+Y+Z, X-Y-Z, X-2Y-Z, X+2Y+Z

X+Y      X-2Y+Z  X+Y-Z
X-Z      X       X+Z
X-Y+Z    X+2Y-Z  X-Y

Terms: X, X+Y, X-Y, X+Z, X-Z, X+Y-Z, X-Y+Z, X+2Y-Z, X-2Y+Z

Terms in common: X, X+Y, X-Y, X+Z, X-Z

Terms not in common: X+Y+Z  VS X+Y-Z  AND X-Y-Z  VS X-Y+Z
X-2Y-Z VS X-2Y+Z AND X+2Y+Z VS X+2Y-Z

I haven't convinced myself that you don't need to iterate over other series of equations. Still thinking on it though.

Update: In fact, I think I can show that your statement "By rotating and reflecting we can make the largest corner be x+y" is not in fact true - consider the following square:

```X+Y   X-Y-Z  X+Z
X-Y+Z X      X+Y-Z
X-Z   X+Y+Z  X-Y
You have X+Y and X+Z both in a corner so you need to make a relationship between them to determine which is the largest value. Also, I noticed that I showed it is possible to have X+Y+Z in a corner and am now convinced that multiple series of equations must be iterated (though you can re-use the values for X, Y and Z).

Cheers - L~R

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Re^5: magic squares
by tilly (Archbishop) on Apr 08, 2009 at 16:08 UTC
It is ugly, but I used the above logic to code up a reasonably efficient solution. On my machine it runs in about 0.01s:
tilly,
Thanks! After having read your code and your proof I see why it wasn't immediately obvious to me. I was stuck looking at it a different way. I stubbornly had to convince myself that my way was in fact correct. After I did, I realized we were saying the same thing. This is the frustrating part - I sometimes can't get out of my own way.

Unfortunately, I now lack all ambition to implement my solution.

I only got interested in the problem because glancing at the commentary of existing posts - a number thought brute forcing was necessary. I had a hunch that this wasn't true.

Cheers - L~R

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