Two methods for finding 3x3 semi-magic squares
of cubes
by Lee Morgenstern, May 2010.
(Second
method corrected by Lee Morgenstern in April 2015, after remarks received
from Tim S. Roberts)
We previously searched for a solution using a list containing values which are the difference of two cubes in three different ways. This was the formulation.
A^3 B^3 C^3
D^3 E^3 F^3
G^3 H^3
I^3
If you just tried finding two triples where the C,E,G values already matched, then you didn't test all the possibilities.
Suppose you had two triples which didn't match
where a /= g, c /= i, e /= k
It could still be a solution if a/g = c/i = e/k. This is because each row can be scaled by a different cube which then forces them to match the pattern.
(am)^3 - (bm)^3 = (cm)^3 - (dm)^3 = (em)^3 -
(fm)^3
(gn)^3 - (hn)^3 = (in)^3 - (jn)^3 = (kn)^3 - (ln)^3
A = jn
B = fm
C = am = gn
D = bm
E =
cm = in
F = ln
G = em = kn
H = hn
I = dm
(I think that the previous search only looked at primitive triples. All triples need to be checked to have a reasonable chance of finding something).
In the following, assume all variables are cubes.
Formulation
A B C
D E F
G H I
A + B = I + F
A + D = I + H
A + G = E + F
A + C = E + H
Find two taxicabs
a + b = c + d
e + f = g + h
such that ag = ce.
Cross multiply so that A and I match.
ae + be = ce + de --> A +
B = I + F
ae + af = ag + ah --> A + D = I + H
Find another two taxicabs
i + j = k + l
m + n = p + q
such that ip = km.
Cross multiply so that A and E match.
im + jm = km + lm --> A +
G = E + F
im + in = ip + iq --> A + C = E + H
Cross multiply so that A, F and H match
aeim + beim = ceim + deim --> A
+ B = I + F
aeim + afim = agim + ahim --> A + D = I +
H
aeim + aejm = aekm + aelm --> A + G = E + F
aeim
+ aein = aeip + aeiq --> A + C = E + H
This requires
al = di
and
eq = hm.
We then have the semi-magic square
A = aeim
B = beim
C = aein
D =
afim
E = aeip = aekm --> ip = km
F = aelm = deim --> al = di
G
= aejm
H = aeiq = ahim --> eq = hm
I = agim = ceim --> ag =
ce
aeim beim aein
afim
aeip=aekm aelm=deim
aejm aeiq=ahim agim=ceim
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