Magic series

A magic series is a set of distinct positive numbers which add up to the magic constant of a magic square and a magic cube, thus potentially making up lines in magic tesseracts.

So, in an n × n magic square using the numbers from 1 to n², a magic series is a set of n distinct numbers adding up to n(n²+1)/2. For n = 2, there are just two magic series, 1+4 and 2+3. The eight magic series when n = 3 all appear in the rows, columns and diagonals of a 3 × 3 magic square.

Maurice Kraitchik gave the number of magic series up to n = 7 in Mathematical Recreations in 1942 (sequence A052456 in OEIS). In 2002, Henry Bottomley extended this up to n = 36 and independently Walter Trump up to n = 32. In 2005, Trump extended this to n = 54 (over 2×10¹¹¹) while Bottomley gave an experimental approximation for the numbers of magic series:

$\frac{1}{\pi} \cdot \sqrt{\frac{3}{e}} \cdot \frac{(e n)^n}{n^3-\frac{3}{5}n^2+\frac{2}{7}n}$

In July 2006, Robert Gerbicz extended this sequence up to n = 150.

External links

• Walter Trump's pages on magic series
• Number of magic series up to order 150