Multimagic series for cubes
See also the multimagic series for squares

As we have seen for the smallest bimagic cube, it may be interesting, in order to try to construct a p-multimagic cube of order-n, to find all the p-multimagic series of order n, that is to say the series of n different integers, from 1 to n³, having the correct magic, bimagic,... until p-multimagic sums (= S1, S2,... Sp):

• S1 = n(n³ + 1)/2
• S2 = n(2n⁶ + 3n³ + 1)/6
• S3 = n(n⁹ + 2n⁶ + n³)/4 = n² S1²
• S4 = n(6n¹² + 15n⁹ + 10n⁶ - 1)/30

The order 3 allows to get bimagic series. Here are the 4 possible bimagic series:

• 3       19      20
• 4       15      23
• 5       13      24
• 8        9       25

So that means that:

1. 3 + 19 + 20 = 4 + 15 + 23 = 5 + 13 + 24 = 8 + 9 + 25 = 42 = S1
2. 3² + 19² + 20² = 4² + 15² + 23² = 5² + 13² + 24² = 8² + 9² + 25² = 770 = S2

For the order 4, there are 8 bimagic series for which the list is given in the Smallest bimagic cube page.

Here is a summary of the number of multimagic series, some of these lists being downloadable as Excel files, from 44Kb to 540Kb each. The numbers of bimagic series of order 9 and 10 were computed by Walter Trump, Germany, in October 2005. The number of trimagic series of order 9 was computed by Gildas Guillemot, France, in December 2006 and later confirmed by Michael Quist, USA, in May 2008.

Number of multimagic series for cubes

Order	Bimagic	Trimagic	Tetramagic
3	4	0	0
4	8	0	0
5	272	0	0
6	25 270	(*)  0	0
7	5 152 529	161	0
8	1 594 825 624	17 218	0
9	651 151 145 259	363 949	(**)  0
10	347 171 191 981 324	(*)  0	0
11	Unknown! Look at the unresolved problems

(*) Trimagic series of order 4k+2 are impossible: it is impossible to have S3 even, with S1 odd and S2 odd.
(**) Short and nice proof by Robert Gerbicz, Hungary, in January 2006. "For the order 9, the tetramagic sum is S4=510,118,152,189==13 mod 16. But (2*x+1)^4==1 mod 16 and (2*x)^4==0 mod 16: it means that at least 13 odd numbers are needed. Impossible with only 9 numbers!"

The bimagic and trimagic series are referenced respectively under the numbers A090653 and A092312 in the Neil Sloane's Encyclopedia of Integer Sequences, AT&T Research.

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