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.

Example on magic squares of order 4 :
Magic constant: S1 = (1+2+3+...+16)/4 = 34
Bimagic constant: S2 = (12+22+32+...+162)/4 = 374
Trimagic constant: S3 = (13+23+33+...+163)/4 = 4624
The sequence  2, 8, 9, 15  is a trimagic series because it fulfills three conditions
  (1)   2  + 8  + 9  + 15  = 34 = S1
  (2)   22 + 82 + 92 + 152 = 374 = S2
  (3)   23 + 83 + 93 + 153 = 4624 = S3

The algorithm used for normal magic series does not work for multimagic series.

Bimagic series

Say Bi(m) is the number of bimagic series of order m and dimension 2 (i.e. series for bimagic squares).

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 gb32-program 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 · 1021 - - -
19 3.33 · 1023 - - -
20 3.29 · 1025 - - -

A bimagic series of order 16 consists of 4k even and (16-4k) odd numbers. Thus the problem can be partitioned into 3 tasks.
Note that each series with 4 even numbers has a complementary series with 12 even numbers (replace all numbers i by 257-i).
  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

Trimagic series

Order Trimagic series
3 0
4 2
5 2
6 0
7 0
8 121
9 126
10 0
11 31 187
12 2 226 896
13   17 265 701
During the 20th 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 order-12 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 order-13 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.

Bimagic series for cubes

Order Bimagic series
for cubes
3 4
4 8
5 272
6 25 270
7 5 152 529
8 1 594 825 624
9 651 151 145 259
10 347 171 191 981 324
 Of course there also are bimagic series for magic cubes.

Christian Boyer (France) calculated the number of bimagic cube series up to order 8 with backtracking method in October 2003.
The above described split sum method of Lorenz Schlangen (Germany) can also be used for cube series.
This enabled me to determine the number of bimagic series for cubes of the orders 9 and 10 in October 2005.
Christian Boyers results could also be confirmed by the new method.
The number of bimagic order-8 cube series can now be calculated in 16 seconds.

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Walter Trump, Nürnberg, Germany, (c) 2005-02-08 (last modified: 2005-11-02)