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 pmultimagic cube of ordern, to find all the pmultimagic series of order n, that is to say the series of n different integers, from 1 to n^{3}, having the correct magic, bimagic,... to pmultimagic sums (= S1, S2,... Sp):
The order 3 allows one to get bimagic series. Here are the 4 possible bimagic series:
So that means that:
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: look at his algorithm (PDF file).
Order 
Bimagic 
Trimagic 
Tetramagic 
Pentamagic 
Hexamagic 
3 
4 
0 
0 
0 
0 
4 
0 
0 
0 
0 

5 
0 
0 
0 
0 

6 
(a) 0 
0 
0 
0 

7 
5152529 
0 
0 
0 

8 
1594825624 
0 
0 
0 

9 
651151145259 
363949 
(b) 0 
0 
0 
10 
347171191981324 
(a) 0 
0 
0 
0 
11 
315035719463520007 
Unknown! (d) 
(e) 0 
0 
0 
12 
333498789992790704850 
Unknown! (d) 
(e) 0 
0 
0 
13 
450285458654002877929960 
Unknown! but > 0 
Unknown! (d) 
(e) 0 
0 
14 
Unknown! (c) ~ 7.4 · 10^{26} 
(a) 0 
0 
0 
0 
15 
Unknown! (c) ~ 1.5 · 10^{30} 
Unknown! (d) 
Unknown!!!!!! 
(e) 0 
0 
16 
Unknown! (c) ~ 3.7 · 10^{33} 
Unknown! but > 0 
Unknown! but > 0 
Unknown! (d) 
(e) 0 
17 
Unknown! (c) ~ 1.1 · 10^{37} 
Unknown! but > 0 
(e) 0 
0 
0 
18 
Unknown! (c) ~ 3.7 · 10^{40} 
(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, AprilMay 2015
(e) Impossibility
proofs by Lee Morgenstern, AprilMay 2015
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 decrease with higher orders. In August 2015, Dirk Kinnaes improved Quist's formula, adding a new term (look also his PDF file given above).
Order  Quist  Kinnaes  True number  Quist error  Kinnaes error 
10  3.607 · 10^{14}  3.693 · 10^{14}  347171191981324  +3.90%  +6.36% 
11  2.971 · 10^{17} 
3.029 · 10^{17} 
315035719463520007  5.68%  3.86% 
12  3.197 · 10^{20} 
3.248 · 10^{20} 
333498789992790704850  4.15%  2.61% 
13  4.365 · 10^{23}  4.424 · 10^{23} 
450285458654002877929960  3.07%  1.75% 
14  7.389 · 10^{26}  7.475 · 10^{26}  ?  ? 

15  1.521 · 10^{30}  1.536 · 10^{30}  ?  ? 

16  3.747 · 10^{33}  3.780 · 10^{33}  ?  ? 

17  1.089 · 10^{37}  1.098 · 10^{37}  ?  ? 

18  3.691 · 10^{40}  3.716 · 10^{40}  ?  ? 

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