# Magic star

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Special arrangement of numbers on a star

Not to be confused with [Star Magic](/source/Star_Magic).

An *n*-pointed **magic star** is a [star polygon](/source/Star_polygon) with [Schläfli symbol](/source/Schl%C3%A4fli_symbol) {*n*/2}[1] in which numbers are placed at each of the *n* [vertices](/source/Vertex_(geometry)) and *n* intersections, such that the four numbers on each line sum to the same [magic constant](/source/Magic_constant).[2] A **normal** magic star contains the [integers](/source/Integer) from 1 to 2*n* with no numbers 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 hexagrams](/source/Magic_hexagram)* (n = 6), *magic heptagrams* (n = 7), etc.

Magic hexagram M = 26 Magic heptagram M = 30 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, ... (sequence [A200720](https://oeis.org/A200720) in the [OEIS](/source/On-Line_Encyclopedia_of_Integer_Sequences)).

## See also

- [Magic square](/source/Magic_square)

## References

1. **[^](#cite_ref-1)** [Weisstein, Eric W.](/source/Eric_W._Weisstein) ["Star Polygon"](https://mathworld.wolfram.com/StarPolygon.html). *[MathWorld](/source/MathWorld)*.

1. **[^](#cite_ref-2)** Staszkow, Ronald (2003-05-01). [*Math Skills: Arithmetic with Introductory Algebra and Geometry*](https://archive.org/details/mathskills00rona). Kendall Hunt. p. [374](https://archive.org/details/mathskills00rona/page/374). [ISBN](/source/ISBN_(identifier)) [9780787292966](https://en.wikipedia.org/wiki/Special:BookSources/9780787292966). magic star math.

1. **[^](#cite_ref-3)** ["Magic Stars Index Page"](http://www.magic-squares.net/magic_stars_index.htm#Introduction). *www.magic-squares.net*. Retrieved 2017-01-14.

## External links

- [Marian Trenkler's Magic Stars](http://math.ku.sk/~trenkler/MagicStars.doc)

- [Gianni Sarcone's Magic Star](http://www.archimedes-lab.org/pzm58b.html)

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v t e Magic polygons Types Magic circle Magic hexagon Magic hexagram Magic square Magic star Magic triangle Related shapes Alphamagic square Antimagic square Geomagic square Heterosquare Pandiagonal magic square Prime reciprocal magic square Most-perfect magic square Higher dimensional shapes Magic cube classes Magic hypercube Magic hyperbeam Classification Associative magic square Pandiagonal magic square Multimagic square Related concepts Latin square Word square Number Scrabble Eight queens puzzle Magic constant Magic graph Magic series

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Adapted from the Wikipedia article [Magic star](https://en.wikipedia.org/wiki/Magic_star) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Magic_star?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.