Trenkler Stars

Type S or Type T	There are two bsically different types of magic stars with 4 numbers in a row .
Weakly and Almost	Some stars are only weakly magic or are almost magic.

Marián Trenkler of Safarik University, Kosice, Slovakia published a paper on Magic Stars.
It is called "Magicke hviezdy" (Magic stars) and appeared in Obzory matematiky, fyziky a informatiky, 51(1998), pages 1-7. (Obzory = horizons (or line of sight) of mathematics, physics and informatics)

The following material is taken from this paper with his permission. Thanks Marian.

Type S is indicated by S_n and has all the numbers external to the pattern (what I call peaks and valleys).
My normal magic stars are all of this type.

Type T stars are so indicated with T_n (n in each case, is the order of the star). Type T stars have numbers on the peaks also, but instead of numbers in the valleys, they are at intersections in the interior of the pattern. There are type T magic stars for all orders greater then six. However, there will not be as many patterns (for a given order) as there are for type S stars. For example, order-7 has only 1 instead of 2 and order-12 has only 3 patterns for Type T but 4 patterns for type S.

Here is an order-6 Type S star (my index # 39) and an order-8 Type T star. The superscript 'M' indicates the star is magic.

There exist stars of all orders greater then four that are almost magic or consist of non-consecutive numbers. They are defined thus. The superscript 'W' indicates the star is weakly magic, the superscript 'A' indicates almost-magic.

Here are two order-5 stars.

The first one is weakly-magic because all five lines sum to the same value, but the numbers used are not consecutive.

The second star is almost-magic because, although it uses consecutive numbers, only three lines sum correctly to 22. One sums to 21 and one sums to 23

Weakly-magic Stars have all lines summing to the same value, but consist of numbers that are not in consecutive order. All order-5 magic stars are of this type. Prime number magic stars are also of this type.

For this example, I have used a prime number magic star!

Almost magic stars use consecutive numbers but have n – 2 lines summing to 4n + 2, and the other two sum to 4n + 1 and 4n + 3. If we attempt to construct an order-5 magic star with numbers 1 to 10, this is the best we can do.
This order-7 star has 5 lines summing to 30, 1 line sums to 29 and 1 line sums to 31

Footnote: Charles W. Trigg, Journal of Recreational Mathematics 29:1, 1998, p 8-11 uses the same term (almost magic) but applies it only to pentagrams (order-5 stars). Also, his definition has 4 lines summing correctly.