12x12 magic squares of cubes.
12x12 magic squares of 4th powers.
12x12 magic squares of 5th powers.
See also the Magic squares of cubes general page

At least one 12x12 magic square of cubes is known: using the 12x12 trimagic square of Walter Trump constructed in 2002, it is easy to obtain a magic square of cubes, directly raising to the third power its integers.

François Labelle, inspired by Lee Morgenstern's 6x6 method, found in April 2010 a method to construct 12x12 semi-magic squares of fourth powers (or of any Nth-powers).

If the two equations (12.1) (12.2) are true:

then this square is a semi-magic square of Nth-powers, with magic sum SN = yz:

Here is his solution of fourth powers, giving the smallest magic sum and 144 distinct integers:

generating the square:

This method can't be applied to 5th powers: because nobody knows a Taxicab(5, 3, 3) number (means a5 + b5 + c5 = d5 + e5 + f5 = g5 + h5 + i5), it will be very difficult to find a solution of the more difficult (12.1) equation which is a Taxicab(5, 3, 4) number!

Open problems:

March 2018, Nicolas Rouanet, France, constructed this 12x12 (and also a 10x10) nearly-magic square of consecutive 4th powers from 0^4 to 143^4:

Nicolas Rouanet remarked, using a reasoning modulo 5, that a 12x12 magic (or semi-magic) square is impossible using consecutive 4th powers from 1^4 to 144^4.

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