magic star

{{Short description|Special arrangement of numbers on a star}}

{{Distinguish|Star Magic}}

An n-pointed magic star is a star polygon with Schläfli symbol {n/2}{{MathWorld |urlname=StarPolygon |title=Star Polygon}} 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.{{Cite book|url=https://archive.org/details/mathskills00rona|url-access=registration|page=[https://archive.org/details/mathskills00rona/page/374 374]|quote=magic star math.|title=Math Skills: Arithmetic with Introductory Algebra and Geometry|last=Staszkow|first=Ronald|date=2003-05-01|publisher=Kendall Hunt|isbn=9780787292966|language=en}} A normal magic star contains the integers from 1 to 2n with no numbers repeated.{{Cite web|url=http://www.magic-squares.net/magic_stars_index.htm#Introduction|title=Magic Stars Index Page|website=www.magic-squares.net|access-date=2017-01-14}} The magic constant of an n-pointed normal magic star is M = 4n + 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 hexagrams (n = 6), magic heptagrams (n = 7), etc.

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valign="bottom" | style="padding: 0 1em" | file:Magic6star-sum26.svg | style="padding: 0 1em" | file:magic7star-sum30.svg | style="padding: 0 1em" | file:magic8star-sum34.svg

align="center" | Magic hexagram M = 26 | align="center" | Magic heptagram M = 30 | align="center" | Magic octagram M = 34

The number of distinct normal magic stars of type {n/2} for n up to 15 is,

: 0, 80, 72, 112, 3014, 10882, 53528, 396930, 2434692, 15278390, 120425006, ... {{OEIS|A200720}}.