The
following text is written by Walter Trump and is coming from his web site
www.trump.de/magicsquares
,Thanks
for his authorization to use his text.
Magic Series  Multimagic series
In multimagic series additionally the sums of certain powers of the numbers have to equal constant values.
There are bimagic, trimagic, tetramagic, ... series for squares, cubes and hypercubes of any dimension.
The algorithm used for normal magic series does not work for multimagic series.
(Download Introduce Of Magic Series )
(Download Magic Series Demostrate Program)
On his famous website http://www.multimagie.com/ (external link) Christian Boyer publishes the following numbers of series for bimagic and trimagic squares. Most numbers were determined by Boyer and me, Bi(12) and Tri(13) were reported by Fredrik Jansson (Finland). I added Bi(13) to Bi(16). These values were approximated by Monte Carlo methods (standard deviation < 0.5%).
Order 
Bimagic series 
Trimagic series 
3 
0 
0 
4 
2 
2 
5 
8 
2 
6 
98 
0 
7 
1 844 
0 
8 
38 039 
121 
9 
949 738 
126 
10 
24 643 236 
0 
11 
947 689 757 
31 187 
12 
45 828 982 764 
2 226 896 
13 
2.16 · 10^{12} 
17 265 701 
14 
1.24 · 10^{14} 
? 
15 
8.15 · 10^{15} 
? 
16 
5.74 · 10^{17} 
? 
In 2002 I used a complete list of trimagic order12 series to find the smallest trimagic square.
This discovery and all about multimagic series is described on the above mentioned site of Christian Boyer.
Walter Trump, Nürnberg, Germany, (c) 20050208 (last modified: 20050216)
