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
[download]

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
[download]

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).

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