An *n*-pointed **magic star** is a star polygon with Schläfli symbol {*n*/2}^{ [1] } in which numbers are placed at each of the *n* vertices and *n* intersections, such that the four numbers on each line sum to the same magic constant.^{ [2] } A **normal** magic star contains the consecutive integers 1 to 2*n*. No numbers are ever repeated.^{ [3] } The magic constant of an *n*-pointed normal magic star is *M* = 4*n* + 2.

No star polygons with fewer than 5 points exist, and the construction of a normal 5-pointed magic star turns out to be impossible. It can be proven that there exists no 4-pointed star that will satisfy the conditions here. The smallest examples of normal magic stars are therefore 6-pointed. Some examples are given below. Notice that for specific values of *n*, the *n*-pointed magic stars are also known as *magic hexagram* etc.

Magic hexagramM = 26 | Magic heptagramM = 30 | Magic octagramM = 34 |

In elementary geometry, a **polytope** is a geometric object with "flat" sides. It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions *n* as an *n*-dimensional polytope or ** n-polytope**. Flat sides mean that the sides of a (

In elementary 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**.

In recreational mathematics and combinatorial design, a **magic square** is a square grid filled with distinct positive integers in the range such that each cell contains a different integer and the sum of the integers in each row, column and diagonal is equal. The sum is called the *magic constant* or *magic sum* of the magic square. A square grid with n cells on each side is said to have *order n*.

In mathematics, a **polygonal number** is a number represented as dots or pebbles arranged in the shape of a regular polygon. The dots are thought of as alphas (units). These are one type of 2-dimensional figurate numbers.

In geometry, a **star polygon** is a type of non-convex polygon. Only the **regular star polygons** have been studied in any depth; star polygons in general appear not to have been formally defined, however certain notable ones can arise through truncation operations on regular simple and star polygons.

The term **figurate number** is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes and different dimensions. The term can mean

In Euclidean geometry, a **regular polygon** is a polygon that is equiangular and equilateral. Regular polygons may be either **convex** or **star**. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon, if the edge length is fixed.

In geometry, a **diagonal** is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word *diagonal* derives from the ancient Greek διαγώνιος *diagonios*, "from angle to angle" ; it was used by both Strabo and Euclid to refer to a line connecting two vertices of a rhombus or cuboid, and later adopted into Latin as *diagonus*.

In mathematics, a ** P-multimagic square** is a magic square that remains magic even if all its numbers are replaced by their

In geometry, the **Schläfli symbol** is a notation of the form {*p*,*q*,*r*,...} that defines regular polytopes and tessellations.

A **pandiagonal magic square** or **panmagic square** is a magic square with the additional property that the broken diagonals, i.e. the diagonals that wrap round at the edges of the square, also 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. In general where is the side length of the square.

A **centered****triangular number** is a centered figurate number that represents a triangle with a dot in the center and all other dots surrounding the center in successive triangular layers. The centered triangular number for *n* is given by the formula

The **centered polygonal numbers** are a class of series of figurate numbers, each formed by a central dot, surrounded by polygonal layers with a constant number of sides. Each side of a polygonal layer contains one dot more than a side in the previous layer, so starting from the second polygonal layer each layer of a centered *k*-gonal number contains *k* more points than the previous layer.

In geometry, the **Heesch number** of a shape is the maximum number of layers of copies of the same shape that can surround it. **Heesch's problem** is the problem of determining the set of numbers that can be Heesch numbers. Both are named for geometer Heinrich Heesch, who found a tile with Heesch number 1 and proposed the more general problem.

In geometry, a **skew polygon** is a polygon whose vertices are not all coplanar. Skew polygons must have at least 4 vertices. The *interior* surface of such a polygon is not uniquely defined.

A **mathematical constant** is a number whose value is fixed by an unambiguous definition, often referred to by a symbol or by mathematicians' names to facilitate using it across multiple mathematical problems. Constants arise in many areas of mathematics, with constants such as e and π occurring in such diverse contexts as geometry, number theory, and calculus.

A **magic hexagram** of order *2* is an arrangement of numbers in a hexagram with triangular cell with *2* cells on each edge, in such a way that the numbers in each row, in all three directions, sum to the same magic constant *M*.

- ↑ Weisstein, Eric W. "Star Polygon".
*MathWorld*. - ↑ Staszkow, Ronald (2003-05-01).
*Math Skills: Arithmetic with Introductory Algebra and Geometry*. Kendall Hunt. p. 374. ISBN 9780787292966.magic star math.

- ↑ "Magic Stars Index Page".
*www.magic-squares.net*. Retrieved 2017-01-14.

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