3x3 Semi-Magic Square of Cubes
Formulation

There is no 3x3 magic square of cubes with distinct integers because of the
arithmetic progressions that are required in any 3x3 magic square.
An arithmetic progression of positive distinct cubes is impossible.
If negative values are allowed, then the middle term must be zero,
in which case there must be three zeroes in the full square.

An open problem is to find a 3x3 semi-magic square of cubes.
Here is a formulation that will enable your computer to find
all possible solutions in a large range of numbers.
It is an example of a multi-phase sort-scan technique.
The first section is a proof of equivalence.
The second section is a suggested computer procedure
that takes advantage of the symmetries in the formulation.

All variables below are assumed to be cubes.

3x3 Semi-Magic Equivalence Theorem
Find two numbers, each of which can be expressed as the
difference of two cubes in three different ways.
If the two expressions have the three cubes C,E,G, in common as follows
C-D = E-I = G-B
C-H = E-A = G-F
then the cubes can be arranged into the following
3x3 semi-magic square of cubes.
A B C
D E F
G H I
Also, all semi-magic squares of cubes are representable by a
solution to the given expressions.

Proof of Necessity
D+E+F = C+F+I ==> D+E = C+I ==> C-D = E-I
B+E+H = G+H+I ==> B+E = G+I ==> E-I = G-B
A+B+C = B+E+H ==> A+C = E+H ==> C-H = E-A
A+D+G = D+E+F ==> A+G = E+F ==> E-A = G-F

Proof of Sufficiency
C-D = E-I ==> C+I = D+E ==> D+E+F = C+F+I
C-H = G-F ==> C+F = G+H ==> C+F+I = G+H+I
E-I = G-B ==> G+I = B+E ==> G+H+I = B+E+H
C-H = E-A ==> E+H = A+C ==> B+E+H = A+B+C
C-D = G-B ==> B+C = D+G ==> A+B+C = A+D+G

Computer Procedure
(1) Make a list of pairs of integers, C and D, and the difference of their cubes (C-D)
for a large range of values for C and D.
(2) Sort the list by (C-D) so that all pairs having the same difference
will be grouped together.
(3) Scan the list for triples of pairs having matching (C-D) values
and make a second list of all possible triples
(C-D) = (E-I) = (G-B), where C < E < G.
For each triple, make sure another triple is listed having
the values (D-C) = (I-E) = (B-G).
[Note: if C < E < G, then D < I < B, so extra permuting is not necessary].
(4) Sort this second list by C,E,G (the initial value of each pair) so that
triples having the same initial values will be grouped together.
(5) Scan the second list for pairs of triples having matching C,E,G values,
but distinct values for all others.
Assign the values to the following variables.
C-D = E-I = G-B
C-H = E-A = G-F
Arrange the values into the following 3x3 square.
A B C
D E F
G H I