| Magic Series |
| Table of Contents |
| Introduction (Easy !) | |
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What are magic series and what can we do with them? | |
| How to count magic series (Interesting !) | |
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How can we calculate the number of magic series for a certain order? | |
| Theory of magic series (pdf) (Impressive !) | |
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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 !) | |
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These numbers have an accuracy of 15 digits. (2005-02-02) | |
| Exact number of magic series up to order 1000 (Incredible !) | |
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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 !) | |
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There are also series for magic objects of higher dimensions. | |
| Formulae for random dimensions (Exciting !) | |
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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 !) | |
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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 |
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Read how magic series can be used to estimate the number of magic squares. | |