An ''n''-pointed magic star is a
star polygon
In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, Decagram (geometry)#Related figures, certain notable ones can ...
with
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines List of regular polytopes and compounds, regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...
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
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 ...
. A normal magic star contains the
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s from 1 to 2''n'' with no numbers repeated.
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'' (n = 6), ''magic heptagrams'' (n = 7), etc.
The number of distinct normal magic stars of type for ''n'' up to 15 is,
: 0, 80, 72, 112, 3014, 10882, 53528, 396930, 2434692, 15278390, 120425006, ... .
See also
*
Magic square
In mathematics, especially History of mathematics, historical and 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 diago ...
References
External links
Marian Trenkler's Magic Stars
Magic figures
Star symbols
{{Magic polygons