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:::Magic Square:::   


  Looking for her for million times

　

  Congratulations


    16th   trimagic square

    24th trimagic square

   40/48th trimagic square

    11th bimagic square

　

    World Record

  
    What's a multimagic square?

    Multimagic formulas

     Magic Series

    Enumeration of magic squares

    Bimagic 8

    Bimagic Square, 9th-order

    Trimagic 12


    Links to Similar Web Sites

The following text is written by Walter Trump and is coming from his web site www.trump.de/magic-squares ,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 · 1012   17 265 701
14 1.24 · 1014 ?
15 8.15 · 1015 ?
16 5.74 · 1017 ?

    In 2002 I used a complete list of trimagic order-12 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) 2005-02-08 (last modified: 2005-02-16) 

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