Magic Series
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 !)
Update 2020-10-28: Lee Morgenstern calculated the number of bimagic series from order 31 to 34.
These series can be used to construct multimagic squares.

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.

Walter Trump, Nürnberg, Germany, (c) 2005-02-08 (last modified: 2020-10-28)