Magic Series 
Multimagic series 
Achille Rilly (France) calculated Bi(8) correctly in 1906 and later Bi(7) before he died in 1909. 
Bi(9) and Bi(10) were found by Christian Boyer (France) in May 2002. 
I calculated Bi(11) in August 2002 with backtracking methods. 
Bi(12) was reported in March 2004 by Fredrik Jansson (Finland). His computer worked about 50 days on this task, using backtracking methods. 
In 2005 Lorenz Schlangen from Solingen, Germany introduced a new method. He splits the numbers of the series into 3 sets and saves the bimagic
sums of each set in a table. With this split sum method Bi(12) can be determined in less than a minute (!)  what a progress. Lorenz Schlangen computed Bi(13), Bi(14) and Bi(15). 
In September 2005 I wrote a gb32program based on Lorenz's ideas. This program could determine Bi(16) within 16 hours. The calculation of Bi(17) took about 4 weeks, using the split sum method. 
In the table below Bi(18) to Bi(20) were approximated by Monte Carlo methods (standard deviation < 0.5%). You can trust these values because previous approximations were close to the true results found later. 
Order 
Number of bimagic series 
First computed  
in ...  by ...  from ...  
3  0  (?) 1892  Michel Frolov  France 
4  2  (?) 1892  Michel Frolov  France 
5  8  ?  ?  ? 
6  98  ?  ?  ? 
7  1 844  before 1909  Achille Rilly  France 
8  38 039  1906  Achille Rilly  France 
9  949 738  May 2002  Christian Boyer  France 
10  24 643 236  May 2002  Christian Boyer  France 
11  947 689 757  August 2002  Walter Trump  Germany 
12  45 828 982 764  March 2004  Fredrik Jansson  Finland 
13  2 151 748 695 931  July 2005  Lorenz Schlangen  Germany 
14  123 821 075 526 032  July 2005  Lorenz Schlangen  Germany 
15  8 131 094 055 190 149  August 2005  Lorenz Schlangen  Germany 
16  573 957 471 153 552 576  September 2005  Walter Trump  Germany 
17  44 010 987 379 157 415 768  October 2005  Walter Trump  Germany 
18  3.65 · 10^{21}       
19  3.33 · 10^{23}       
20  3.29 · 10^{25}       
a)  Bi(16, 0 even) = Bi(16, 16 even) =  44 103 659 723 445 
b)  Bi(16, 4 even) = Bi(16, 12 even) =  57 725 723 619 430 429 
c)  Bi(16, 8 even) =  458 417 816 595 244 828 
Bi(16, 4 even) =  57 725 723 619 430 429  
Bi(16, 0 even) =  44 103 659 723 445  
  
Sum of all Bi(16,4k even) = Bi(16) =  573 957 471 153 552 576 

During the 20^{th} century the numbers of trimagic series of the orders 8, 9 and 11 were published incorrectly in magazines and books.
Christian Boyer was the first to calculate these numbers accurately from 2001 (order 8) to May 2002 (orders 9 and 11). In 2002 I determined all trimagic series of order 12 in order to construct a trimagic order12 square, the smallest possible trimagic square. This discovery and all about multimagic series is described at the famous website www.multimagie.com (external link) of Christian Boyer. In 2004 Fredrik Jansson from Finland calculated the number of trimagic order13 series. He wants to construct a trimagic square of order 13. A really difficult task. As far as I know the number Tri(13) was not confirmed by anybody else until now. 

Of course there also are bimagic series for magic cubes.

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