|Table of Contents|
|Introduction (Easy !)|
|What are magic series and what can we do with them?|
|How to count magic series (Interesting !)|
How can we calculate the number of magic series for a certain order?|
|Theory of magic series (pdf) (Impressive !)|
A mathematical definition and an iterative algorithm for counting magic series.|
Note that I wrote this in September 2002 without knowing that
Henry Bottomley (Great Britain) made a similar approach some months before.
Have a look at his excellent partition calculator (external link).
Robert Gerbicz (Hungary) presented a new algorithm in April 2006. It is faster and needs less memory.
Visit his homepage and read about his C-program (external link).
Dirk Kinnaes (Belgium) found a completely different algorithm in March 2013. It does not use recurrence
relations and can even handle order m = 1000. Read the description of Kinnaes'-algorithm.
|Number of Series up to order 100 (Old but useful !)|
|These numbers have an accuracy of 15 digits. (2005-02-02)|
|Exact number of magic series up to order 1000 (Incredible !)|
The number of magic order-50 series has got more than 100 digits.|
Breakthrough in 2006: Robert Gerbicz (Hungary) extended the table dramatically up to order 150.
Breakthrough in 2013: Dirk Kinnaes (Belgium) calculated N(200) ... N(1000)
Update 2019-10-23: Yukimasa Sugizaki (Japan) calculated many further numbers up to N(3000).
Update 2019-11-12: Katsuhiro Endo and Yukimasa Sugizaki independently calculated N(4000).
|Magic series of cubes and hypercubes (Magic3 !)|
There are also series for magic objects of higher dimensions.|
|Formulae for random dimensions (Exciting !)|
Impressive strategies to enumerate magic series in various dimensions.|
2007: Exact formulae found.
2007: Sequence for first coefficients found.
2013: First coefficients proved mathematically by Dirk Kinnaes.
|Multimagic series (Important !)|
These series can be used to construct multimagic squares and cubes.|
Read more on Christian Boyer's famous site www.multimagie.com (external link).
|Link to other pages:||Number of magic squares of higher orders|
Read how magic series can be used to estimate the number of magic squares.|