A magic series is a set of distinct positive

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s which add up to the

magic constant The magic constant or magic sum of a magic square is the sum of numbers in any row, column, or diagonal of the magic square. For example, the magic square shown below has a magic constant of 15. For a normal magic square of order ''n'' – that is ...

of a

magic square In recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The 'order' of the magic square is the number ...

and a

magic cube In mathematics, a magic cube is the dimension, 3-dimensional equivalent of a magic square, that is, a collection of integers arranged in an ''n'' × ''n'' × ''n'' pattern such that the sums of the numbers on each row, ...

, thus potentially making up lines in

magic tesseract In mathematics, a magic hypercube is the ''k''-dimensional generalization of magic squares and magic cubes, that is, an ''n'' × ''n'' × ''n'' × ... × ''n'' array of integers such that the sums of the numbers on each pillar (along any axis) a ...

s. So, in an ''n'' × ''n'' magic square using the numbers from 1 to ''n''², a magic series is a set of ''n'' distinct numbers adding up to ''n''(''n''² + 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 Maurice Borisovich Kraitchik (21 April 1882 – 19 August 1957) was a Belgian mathematician and populariser. His main interests were the theory of numbers and recreational mathematics. He was born to a Jewish family in Minsk. He wrote seve ...

gave the number of magic series up to ''n'' = 7 in ''Mathematical Recreations'' in 1942 . In 2002,

Henry Bottomley Henry may refer to: People *Henry (given name) *Henry (surname) * Henry Lau, Canadian singer and musician who performs under the mononym Henry Royalty * Portuguese royalty ** King-Cardinal Henry, King of Portugal ** Henry, Count of Portugal ...

extended this up to ''n'' = 36 and independently

Walter Trump Walter Trump (born 1952 or 1953 ) is a German mathematician and retired high school teacher. He is known for his work in recreational mathematics. He has made contributions working on both the square packing problem and the magic tile problem. In ...

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 \cdot \sqrt \cdot \frac

In July 2006,

Robert Gerbicz The name Robert is an ancient Germanic given name, from Proto-Germanic "fame" and "bright" (''Hrōþiberhtaz''). Compare Old Dutch ''Robrecht'' and Old High German ''Hrodebert'' (a compound of '' Hruod'' ( non, Hróðr) "fame, glory, honou ...

extended this sequence up to ''n'' = 150. In 2013 Dirk Kinnaes was able to exploit his insight that the magic series could be related to the volume of a

polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...

. Trump used this new approach to extend the sequence up to ''n'' = 1000.Walter Trump http://www.trump.de/magic-squares/ Mike Quist showed that the exact second-order count has a multiplicative factor of

\tfrac\!\left(1+\tfrac+\tfrac+\cdots\right)

equivalent to a denominator of

n^3-\tfracn^2+\left(\tfrac + \tfrac \right)\!n+ \cdots.

Richard Schroeppel Richard C. Schroeppel (born 1948) is an American mathematician born in Illinois. His research has included magic squares, elliptic curves, and cryptography. In 1964, Schroeppel won first place in the United States among over 225,000 high school st ...

in 1973 published the complete enumeration of the order 5 magic squares at 275,305,224. This recent magic series work gives hope that the relationship between the magic series and the magic square may provide an exact count for order 6 or order 7 magic squares. Consider an intermediate structure that lies in complexity between the magic series and the magic square. It might be described as an amalgamation of 4 magic series that have only one unique common integer. This structure forms the two major diagonals and the central row and column for an odd order magic square. Building blocks such as these could be the way forward.

References

External links

Walter Trump's pages on magic series

Number of magic series up to order 150
* {{Magic polygons Magic squares