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 *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 gave the number of magic series up to *n* = 7 in *Mathematical Recreations* in 1942 (sequence A052456 in the 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 2×10^{111}) while Bottomley gave an experimental approximation for the numbers of magic series:

In July 2006, Robert Gerbicz 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. Trump used this new approach to extend the sequence up to *n* = 1000.^{ [1] }

Mike Quist showed that the exact second-order count has a multiplicative factor of equivalent to a denominator of ^{ [2] }

Richard Schroeppel in 1973 published the complete enumeration of the order 5 magic square 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.

In mathematics, the **Bernoulli numbers***B*_{n} are a sequence of rational numbers which occur frequently in number theory. The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of *m*-th powers of the first *n* positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function.

In mathematics, a **series** is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures, through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance.

In geometry, a **polygon** is a plane figure that is described by a finite number of straight line segments connected to form a closed polygonal chain or *polygonal circuit*. The solid plane region, the bounding circuit, or the two together, may be called a **polygon**.

The **Riemann zeta function** or **Euler–Riemann zeta function**, *ζ*(*s*), is a function of a complex variable *s* that analytically continues the sum of the Dirichlet series

In mathematics, the **Taylor series** of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor's series are named after Brook Taylor who introduced them in 1715.

In number theory, **Euler's totient function** counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as *φ*(*n*) or *ϕ*(*n*), and may also be called **Euler's phi function**. In other words, it is the number of integers k in the range 1 ≤ *k* ≤ *n* for which the greatest common divisor gcd(*n*, *k*) is equal to 1. The integers k of this form are sometimes referred to as totatives of n.

The **Euler–Mascheroni constant** is a mathematical constant recurring in analysis and number theory, usually denoted by the lowercase Greek letter gamma.

In mathematics, the n-th **harmonic number** is the sum of the reciprocals of the first n natural numbers:

In mathematics, an integral polytope has an associated **Ehrhart polynomial** that encodes the relationship between the volume of a polytope and the number of integer points the polytope contains. The theory of Ehrhart polynomials can be seen as a higher-dimensional generalization of Pick's theorem in the Euclidean plane.

The **square root of 2**, or the (1/2)th power of 2, written in mathematics as √2 or 2^{1⁄2}, is the positive algebraic number that, when multiplied by itself, equals the number 2. Technically, it is called the **principal square root of 2**, to distinguish it from the negative number with the same property.

In mathematics, the **double factorial** or **semifactorial** of a number n is the product of all the integers from 1 up to n that have the same parity as n. That is,

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. In general where is the side length of the square.

**Eisenstein series**, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be generalized in the theory of automorphic forms.

In number theory, **Sylvester's sequence** is an integer sequence in which each term of the sequence is the product of the previous terms, plus one. The first few terms of the sequence are

In geodesy, a **meridian arc** measurement is the distance between two points with the same longitude, i.e., a segment of a meridian curve or its length. Two or more such determinations at different locations then specify the shape of the reference ellipsoid which best approximates the shape of the geoid. This process is called the determination of the figure of the Earth. The earliest determinations of the size of a spherical Earth required a single arc. The latest determinations use astro-geodetic measurements and the methods of satellite geodesy to determine the reference ellipsoids.

The Rubik's Cube is the original and best known of the three-dimensional sequential move puzzles. There have been many virtual implementations of this puzzle in software. It is a natural extension to create **sequential move puzzles in more than three dimensions**. Although no such puzzle could ever be physically constructed, the rules of how they operate are quite rigorously defined mathematically and are analogous to the rules found in three-dimensional geometry. Hence, they can be simulated by software. As with the mechanical sequential move puzzles, there are records for solvers, although not yet the same degree of competitive organisation.

In mathematics, the **multiple zeta functions** are generalisations of the Riemann zeta function, defined by

In convex geometry, the **Mahler volume** of a centrally symmetric convex body is a dimensionless quantity that is associated with the body and is invariant under linear transformations. It is named after German-English mathematician Kurt Mahler. It is known that the shapes with the largest possible Mahler volume are the balls and solid ellipsoids; this is now known as the **Blaschke–Santaló inequality**. The still-unsolved **Mahler conjecture** states that the minimum possible Mahler volume is attained by a hypercube.

* Ganita Kaumudi* is a treatise on mathematics written by Indian mathematician Narayana Pandita in 1356. It was an arithmetical treatise alongside the other algebraic treatise called "Bijganita Vatamsa" by Narayana Pandit. It was written as a commentary on the

In combinatorial mathematics, probability, and computer science, in the **longest alternating subsequence** problem, one wants to find a subsequence of a given sequence in which the elements are in alternating order, and in which the sequence is as long as possible.

- Walter Trump's pages on magic series
- Number of magic series up to order 150
- De Loera, Jesús A.; Kim, Edward D. (2013),
*Combinatorics and Geometry of Transportation Polytopes: An Update*, arXiv: 1307.0124 , Bibcode:2013arXiv1307.0124D

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