Contents
Introduction to Order5 
Cell designations & some of the features of Order5
magic squares. 
Definitions 
Review magic star definitions and general features.
This is a deferent page. Use Back button to return. 
Some Examples 
Three patterns with points summing to 15, 30 and 45. 
A Second Series 
Order5 magic star uses the numbers from 1 to 12
with 7 and 11 or 2 and
6 not used. i.e. it is not a pure magic star. The magic constant is
24 or 28 respectively. 
Partner Solutions 
Equivalent to complement pairs, but for series of
nonconsecutive numbers. 
Point Totals 
The 12 point totals in each series form a pattern. 
Basic Solutions 
The 12 basic solutions with S = 24. Also 12 basic
solutions for S = 28. 
Relationship to Treeplanting
Problems 
A magic star pattern is basically a graph in the
form of a polygram. It consists of lines and has numbers at the
nodes. Here are the other 5 patterns with 10 numbers in 10 lines of
4. 
Almost Magic 
Pentagrams with numbers 1 to 10, have only four
lines correct (Trigg), or
generalized for all orders (Trenkler). 
Antimagic Pentagrams 
The opposite to Magic Stars. These have every line
sum different. 
Introduction to
Order5
The 5point star, often referred to as a
pentagram is normally included in a selection of magic stars.
However, it is not a pure magic star because it cannot be
formed with the ten numbers from 1 to 10. The lowest one possible is
composed of the numbers from 1 to 12 with 7 and 11 omitted. The
magic Sum (S) is 24.
The minimal series
There are 12 basic solutions with S=24 and using series 1 to 12
with 7 and 11 omitted. Each solution has four rotations and five
reflections, making a total of 120 variations. The complements of
these solutions are solutions of another series with S = 28.
The point totals vary from 15 to 45 with one solution having the
points summing to 24, the constant S. 

Some Examples
A Second Series
Another set of 12 solutions is obtainable from the same
series of 1 to 12, but leaving out the 2 and 6. The constant (S) for this
series is (70/2) * 5 = 28. As mentioned above, all solutions of this set
are compliments of the 12 solutions of the first set where S = 24.
Series with 1 and 7 missing or 3 and 5 missing also sum to 70
but no solutions exist.
The series from 1 to 11 with 6 missing gives a sum of 60, permitting S = 24, but
no solutions exist.
S = 2/5 of the sum of the series, so regardless of what series
is used, S must be an even
number. Any S (greater then 24) may be obtained by modifying a basic solution in
the following manner:

add 1 to each point or valley will give an S of 26

add 2 to each point or valley will give an S of 28

add 1 to each point and valley will give an S of 28

etc.
Partner Solutions
Any solution for S = 24 may be converted to a partner
solution by the following algorithm, which moves the points to the valleys
and the valleys to the points.
a and e
exchange places
b moves to i, c
moves to j, d moves to c,
f moves to d
g moves to b, h
moves to g, i moves to h,
j moves to f
The resulting solution, while equivalent, will however, be a
rotation and/or a reflection of the basic solution.
The six pairs of solutions are: #1 and #8, #2 and # 7 ( no rotation or
reflection required), # 3 and #9, # 4 and #11, # 5 and # 10, # 6 and # 12.
Point Totals
Point totals for minimal series (no 7 and 11):
15 21 24 27 30 33 36 39 45
27 30 33 Note that all are mod 3 = 0
Point totals with no 2 and 6:
20 26 29 32 35 38 41 44 50
32 35 38 Note that all are mod 3 = 2
As 30 and 35, the middle number in each series of totals, are
each onehalf of the respective series sums, the valleys in each case also sum
to 30 and 35 respectively.
Minimal Basic Solutions
for S = 24
Point pcomp
# A b c D e f G h I J Point pair # Remarks
1 1 3 8 12 2 4 6 10 5 9 33 1
2 1 3 12 8 2 9 5 10 6 4 24 2 Points are magic!
3 1 4 9 10 5 3 6 12 2 8 27 3
4 1 4 10 9 5 8 2 12 6 3 21 4
5 1 8 3 12 6 4 2 9 5 10 30 5 Valleys also 30
6 1 8 12 3 6 10 5 9 2 4 15 6 Points are lowest #'s
7 2 5 8 9 1 4 10 6 3 12 36 2
8 2 5 9 8 1 12 3 6 10 4 27 1
9 3 1 8 12 2 6 4 10 9 5 33 3
10 3 1 12 8 2 5 9 10 4 6 30 5 Valleys also 30
11 4 2 6 12 3 1 8 9 5 10 39 4
12 6 2 4 12 1 3 8 5 9 10 45 6 Points are highest #'s
Minimal Basic Solutions for S = 28
Point pcomp
# A b c D e f G h I J Point pair # Remarks
1 1 5 10 12 3 9 4 11 8 7 32 1
2 1 5 12 10 3 7 8 11 4 9 32 2
3 1 7 9 11 4 5 8 10 3 12 35 3 Valleys also 35
4 1 7 11 9 4 12 3 10 8 5 26 4
5 1 9 7 11 8 5 4 12 3 10 29 5
6 1 9 11 7 8 10 3 12 4 5 20 6 Pts.are lowest numbers
7 3 4 9 12 1 10 5 8 11 7 38 2
8 3 4 12 9 1 7 11 8 5 10 38 1
9 4 3 9 12 1 5 10 8 7 11 44 4
10 4 3 12 9 1 11 7 8 10 5 35 3 Valleys also 35
11 5 1 10 12 3 4 9 11 7 8 41 5
12 8 3 7 10 1 5 12 4 9 11 50 6 Points are highest #'s
The solutions of the second series are complements (after normalizing) of solutions in the first series.
Relationship to Treeplanting
Problems
A popular classification of recreational mathematics
problems are known as treeplanting problems. The problem specifies
how many trees, how many trees per row, and how many rows. This
illustration shows the six basic patterns for 10 objects in 5 rows
of 4, using numbers instead of trees. In each case, the hypothetical
trees have been replaced with 10 of the first 12 integers (there are
no solutions using the numbers 1 to 10). The first
pattern is the magic 5pointed star, the other five maintain the
arrangement of numbers so each line sums correctly.
In each case the 7 and 11 are not used. The sum per
line is 24. If we omit the 2 and the 6 instead of the 7 and 11, the
sum per line is 28. In both cases, there are 12 basic solutions. 

Almost Magic Pentagrams
A magic pentagram, we now know, must have 5 lines summing to an
equal value. Charles Trigg [1]
calls a pentagram with only 4 equal lines but constructed with the
consecutive numbers from 1 to 10, an almost magic pentagram.
The numbers from 1 to 10 sum to 55, and as each appears in two
lines the total for the five lines is 110. If we assume only 4 lines
are correct, then 4x + y = 110.
There are six solutions to this equation. (x,y) = (25, 10), (24,
14), (23, 18), (21, 26), (20, 30), (and (19, 34). These form into 3
complement pairs, and it turns out only two of the pairs can form
valid solutions.
There are seven basic solutions as shown below.
The numbers are placed in the corresponding standard positions shown
to the right.


A B C D E F G H I J X Y
8 2 3 1 10 4 9 7 6 5 24 14
7 2 4 1 5 8 10 3 9 6 24 14
10 2 5 1 7 6 9 4 8 3 27 18
9 2 6 1 5 7 10 3 8 4 27 18
10 3 4 1 9 6 7 5 8 2 27 18
7 4 6 1 10 3 9 8 2 5 27 18
4 3 9 2 6 10 5 8 7 1 27 18


A. is solution number 2 (from the above table) x = 24, y = 14
B. is the complement of A x = 20, y =
30
C. is solution number 3 (from the above table) x = 23, y = 18
D. is the complement of C x = 21, y =
26
Each of the seven basic solutions above, and their complement solutions
can be changed into another eleven solutions by performing the following
transformation (on the basic solution).
Therefore, there are 7 x 2 x 12 = 168 solutions without counting
rotations or reflections.
In each case, line y is the first line of the pattern.
[1] See Charles W. Trigg, Almost
Magic Pentagrams, Journal of Recreational Mathematics 29:1, p. 811, 1998,
for more information.
Footnote: Trenkler almostmagic stars

Marián Trenkler (see Trenkler
Stars) has independently coined the phrase almostmagic,
but generalizes it for all orders of stars.
His definition:
If there are numbers 1, 2, …, 2n located in a star S_{n}
( or T_{n}) so that the sum on n – 2 lines is
4n + 2, on the others 4n + 1 and 4n + 3, we
call it an almostmagic star.
NOTE that by his definition, the order5 almostmagic star has
only 3 lines summing correctly. Trigg’s order5 (the only order he
defines) has 4 lines summing the same. 
Antimagic Pentagrams
Antimagic pentagrams are order5 stars using the
consecutive numbers from 1 to 10. Each of the five lines must sum to
a different value.
Here we show two of many such arrangements. In both these cases
the line sums are consecutive (else they would be referred to as
heteromagic).
Charles Trigg [1] reported
that there are 2208 distinct solutions with consecutive line sums
for the order 5 antimagic pentagram.
(1) Charles Trigg Journal of Recreational
Mathematics, vol.10(3), 197778 

