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Let x and y represent the numbers in the upper right and lower left corners respectively. This forces the array ...
x+y-14 2y-9 x
37-2y 23-y 9
y x-y+9 14
So (x+y-14)+(2y-9)+(x) = x+23 --> x=46-3y. The array can be written solely in terms of y as follows ...
32-2y 2y-9 46-3y
37-2y 23-y 9
y 55-4y 14
Now (2y-9)>0 and (55-4y)>0 --> 4 < y < 14. Since array values are distinct, y cannot equal 9 and 11. So y = 5,6,7,8,10,12, and 13 only. Also, since the maximum value in a 3 by 3 magic square with distinct numbers cannot be in the center nor the corners, we need only consider maximizing (2y-9), (37-2y), and (55-4y). Substituting y=5 into (55-4y) yields a maximum value of 35 for any number in the array.
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