Searches on 3x3 semi-magic squares of cubes
by Lee Morgenstern, January 2013.

There is no 3x3 Semi-Magic Square of Distinct Positive Cubes with all entries under (10^6)^3.

Also, there is no 3x3 semi-magic square using a list of all primitive taxicab(2) solutions with entries under (10^6)^3 that are twice-scaled up to entries under (10^24)^3 as follows.

A^3 B^3 C^3
D^3 E^3 F^3
G^3 H^3 I^3

[1] A^3 + G^3 = E^3 + F^3
[2] A^3 + C^3 = E^3 + H^3
[3] A^3 + B^3 = I^3 + F^3
[4] A^3 + D^3 = I^3 + H^3

From two pairs of taxicab solutions

a^3 + b^3 = c^3 + d^3
e^3 + f^3 = g^3 + h^3
where a/c = e/g

and

i^3 + j^3 = k^3 + l^3
m^3 + n^3 = p^3 + q^3
where i/k = m/p

they can be scaled to

(ae)^3 + (be)^3 = (ce)^3 + (de)^3
(ae)^3 + (af)^3 = (ag)^3 + (ah)^3
where ce = ag, satisfying [1] and [2]

and

(im)^3 + (jm)^3 = (km)^3 + (lm)^3
(im)^3 + (in)^3 = (ip)^3 + (iq)^3
where km = ip, satisfying [3] and [4]

and then if

a/d = i/l  and  e/h = m/q

they can be scaled again to

(aeim)^3 + (beim)^3 = (beim)^3 + (deim)^3
(aeim)^3 + (afim)^3 = (agim)^3 + (ahim)^3
(aeim)^3 + (aejm)^3 = (aekm)^3 + (aelm)^3
(aeim)^3 + (aein)^3 = (aeip)^3 + (aeiq)^3
where beim = agim,  aekm = aeip,
deim = aelm,  ahim = aeiq,
satisfying [1],[2],[3],[4].

If the scaling values, a,e,i,m, have no factors in common, then entries could be as large as (10^24)^3.