Magic 7-Square and 2-Negative
by Lee Morgenstern, December 2011.

Regarding the enigma that asks for a 3x3 magic square with 7 square entries, how to deal with the 2 non-square entries if they are negative?
For example, any 3x3 magic square can be put into the following form with a and b positive. Note that c-a-b is the smallest entry and if a + b > c, the smallest entry will be negative.

  c+a  c-a-b   c+b
c-a+b   c    c-b+a
  c-b  c+a+b   c-a

Suppose that the following 7-square solution was found with the smallest entry negative.  Note that this configuration is similar to the one known 7-square solution.

d^2   -E   f^2
g^2   h^2  i^2
j^2   k^2   L

Is it possible to rearrange the square entries so that the non-squares are both positive?
Is it possible to modify the values and get another 7-square solution with all entries positive?
I don't see how in this configuration.


In some configurations, it is possible to rearrange the square entries so that the non-square entries have different values and become positive.
For example, this can be done with a tilted magic hourglass because both the smallest and largest entries are non-squares.

d^2  -E   f^2
g^2  h^2  i^2
j^2   K   l^2

We can make a regular magic hourglass using the same squares, but with M and N different from -E and K and both positive.
The following assumes f is larger than d.  Since j^2 is positive and is the smallest entry, M and N must both be positive.

d^2  j^2  g^2
 M   h^2   N
i^2  f^2  l^2

Some 6-square configurations have many ways of rearranging squares to get different values for the other 3 entries.
But 7-square configurations are more limited.

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