Magic Stars are normally considered to be
figures of 4 numbers in a row that add to the magic constant. The
magic hexagram is such a pattern, and consists of the numbers 1 to
12 arranged in 6 lines of 4 numbers such that each line sums to the
constant S. The sum of the series 1 to 12 is 78, there are 6 lines
with each cell appearing in 2 lines, so the magic constant (S) is (2
x 78)/6 = 26. [1]
In 1989, Harold
Reiter and David Richie considered an elaboration of this figure
[2].
Suppose 3 additional lines were added to the hexagram (from point to
opposite point). If a number (in the series 1 to 19) is placed at
each point of intersection, there will be 5 numbers on each of the 9
lines. Can these numbers be arranged in such a manner that all 9
lines total to the same value?
They found that
the answer was affirmative. Their algorithm and the resulting
computer search program found all possible solutions. There are
actually 9 separate magic constants, ranging from 46 to 54.
This search
problem was made manageable by using several shortcuts.

Search for
solutions with the magic constant from 46 to 50 only. Solutions
for S= 51 to 54 are obtained from these by complementing the
numbers (i.e. subtract each number from 20). When S=50 the
complement is another solution with S=50.

Consider only
a standard position solution when searching. No need to consider
the other 11 rotations or reflections (which are considered
equivalent in magic objects).
Their search
involved only 18,264,704 checks instead of the 19! that would have
been required if all combinations were checked.
The number of
basic solutions are:
Sum 
Solutions 
46 
91 
47 
284 
48 
377 
49 
888 
50 
1100 
51 
888 
52 
377 
53 
284 
54 
91 
Total 
4380 
[1]
Order6 Magic Stars
[2] Harold Reiter and David Richie , A Complete Solution to the
Magic Hexagon Problem, College Mathematics Journal, Vol. 20:4,
1989, pages 307316. 