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)

Here is a way of searching for a 3x3 semi-magic square of cubes that extends the entries to twice the number of digits.

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.

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.

Suppose gcd(m,n) = 1 and n/m = a/g = c/i = e/k, then the smallest scaling would be
Since am = gn, cm = in, em = kn, we have a solution.
If the list of triples contained cubes of 6-digit numbers, then using the scaling method above could potentially find a solution using the cubes of 12-digit numbers.

(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).

Here is yet another way of searching for a 3x3 semi-magic square of cubes that might be better than my scaling-triple method. It involves more scalings so this can produce solutions with even larger entries than ever before.

In the following, assume all variables are cubes.


Find two taxicabs

such that ag = ce.

Cross multiply so that A and I match.

Find another two taxicabs

such that ip = km.

Cross multiply so that A and E match.

Cross multiply so that A, F and H match

This requires

We then have the semi-magic square

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