Multimagic series for cubes

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,... to 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
• S5 = (3n2S22 - S3)/2

The order 3 allows one 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 on 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 orders 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. The numbers of bimagic series of orders 11 and 12 were computed in May and June 2013 by Walter Trump, using a program written by Lee Morgenstern, USA. In August 2015, Dirk Kinnaes, Belgium, confirmed the previously known numbers of bimagic series, and computed the number of bimagic series of order 13: see his algorithm (PDF file).

 Order Bimagic Trimagic Tetramagic Pentamagic Hexamagic 3 4 0 0 0 0 4 8 0 0 0 0 5 272 0 0 0 0 6 25270 (a)  0 0 0 0 7 5152529 161 0 0 0 8 1594825624 17218 0 0 0 9 651151145259 363949 (b)  0 0 0 10 347171191981324 (a)  0 0 0 0 11 315035719463520007 Unknown! (d) (f) > 5.4 · 109 (e)  0 0 0 12 333498789992790704850 Unknown! (d) (f) > 2.5 · 1012 (e)  0 0 0 13 450285458654002877929960 Unknown! (d) (f) > 1.6 · 1015 Unknown! (d) (e)  0 0 14 Unknown! (c) ~ 7.4 · 1026 (a)  0 0 0 0 15 Unknown! (c) ~ 1.5 · 1030 Unknown! (d) (f) > 1.4 · 1021 Unknown!!!!!! (e)  0 0 16 Unknown! (c) ~ 3.7 · 1033 Unknown! (d) (f) > 1.9 · 1024 Unknown! (d) Unknown! (d) (e)  0 17 Unknown! (c) ~ 1.1 · 1037 Unknown! (d) (f) > 3.0 · 1027 (e) 0 0 0 18 Unknown! (c) ~ 3.7 · 1040 (a)  0 0 0 0

(a) Trimagic series of order 4k+2 are impossible: it is impossible to have S3 even, with S1 odd and S2 odd.
(b) 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!"
(c) Estimations of bimagic series by Michael Quist, June 2013.
(d) Unknown numbers of series, but existing series, examples found by Lee Morgenstern, April-May 2015
(e) Impossibility proofs by Lee Morgenstern, April-May 2015
(f) Estimations of trimagic series by Dirk Kinnaes, March 2017

In June 2013, Michael Quist wrote a paper estimating the numbers of magic and multimagic series, including bimagic series for cubes of order N. See http://arxiv.org/abs/1306.0616. Here is his formula, and the obtained numeric values for 10 ≤ N ≤ 18. The error decreases with higher orders. In August 2015, Dirk Kinnaes improved Quist's formula, adding a new term (see also his PDF file given above).

 Order Quistestimated number Kinnaesestimated number True number Quist error Kinnaes error 10 3.607 · 1014 3.693 · 1014 347171191981324 +3.90% +6.36% 11 2.971 · 1017 3.029 · 1017 315035719463520007 -5.68% -3.86% 12 3.197 · 1020 3.248 · 1020 333498789992790704850 -4.15% -2.61% 13 4.365 · 1023 4.424 · 1023 450285458654002877929960 -3.07% -1.75% 14 7.389 · 1026 7.475 · 1026 ? ? ? 15 1.521 · 1030 1.536 · 1030 ? ? ? 16 3.747 · 1033 3.780 · 1033 ? ? ? 17 1.089 · 1037 1.098 · 1037 ? ? ? 18 3.691 · 1040 3.716 · 1040 ? ? ?

In March 2017, Dirk Kinnaes wrote a paper "Estimating the number of K-multimagic hypercube series" (PDF file) including a formula estimating the numbers of trimagic series for cubes of order N. Here is his formula, and the obtained numeric values for 7 ≤ N ≤ 15. The error will decrease with higher orders, we should expect much smaller errors for orders > 10.

 Order Kinnaesestimated number True number Kinnaes error 7 7.0 161 / 23.1 8 544.0 17218 / 31.6 9 74781.2 363949 / 4.9 10 1.65 · 107 0 --- 11 5.42 · 109 ? ? 12 2.52 · 1012 ? ? 13 1.60 · 1015 ? ? 14 1.33 · 1018 0 --- 15 1.41 · 1021 ? ?

The bimagic and trimagic series are referenced respectively under the numbers A090653 and A092312 in the On-Line Encyclopedia of Integer Sequences, OEIS Foundation.