Contents
Introduction 
Isolike, panmagic and other defintions. 
Order5 Isomorphic magic star 
The star that started it all (reproduced from my
Unusual Magic Stars page).
This is the only order8 magic star that is fully isomorphic
to an order5 magic square. 
Regular order  5 
This star is isomorphic to a regular order5 magic
square. But only 10 lines. 
Order7 Panmagic star 
This star is a transformation of an order7 panmagic
square. 
Order9 Isolike magic star 
This star is a transformation of an order9 panmagic
and diammagic square. However, it is incomplete, because the main
diagonals would require duplicate numbers. 
Order9 Butterfly stars 
An order9 panmagic square transforms to two
order12 magic stars, one with an embedded order8 magic star. 
Order13 Isolike magic star 
This star is a transformation of an order13
diammagic and plusmagic square. 
Comparison of features 
Compares the characteristics of Isolike, Isomorphic
and Panmagic stars. 
Conclusions and Questions 
A wrapup, more about plusmagic and diammagic, food
for thought. 
Summary 
Still more wrapup. 
Introduction
What are PanMagic Stars?
On April 30, 1999, I received an email from Aale de Winkel commenting on the
Fiveinarow(8) magic star located on my Unusual Magic Squares page. This star
(which is reproduced below) is obtained by a transformation of an order5
pandiagonal magic square into an order8B magic star..
This message was the first of an intense exchange of communications between us
during the month of May, as we jointly investigated the properties of this star
configuration.
Aale's investigations involve using the pandiagonals to help form an order8B
magic star. Hence the name Panmagic stars. We originally assumed that magic
stars of this type are isomorphic only to pandiagonal magic squares (not regular
magic squares).
This investigation also resulted in the discovery (by Aale) of magic squares
with various patterns beside the usual rows, columns and diagonals.
What are Isolike Magic Stars?
My emphasis has been on the formation of type order8B magic stars using only
numbers from the magic square that will contribute to forming complete magic
lines.
My efforts (and the examples here) involve the embedded "plus", "diamond",
"cross" or "ring" patterns and do not necessarily require the magic square be
pandiagonal. I call this type of square a Quadrant Magic Square.
 Isolike magic stars are similar to an order8, pattern B type magic star.
However, the star is referred to as ordern where n is the
order of the magic square it is derived from.
 They also have 12 correct lines (instead of 8). If they are incomplete,
they do not have the 2 main diagonals, so only 10 lines are correct .
 They have n numbers per line where n is the order of the
magic square the star is transformed from.
My special thanks to Aale de Winkel for the use of his panmagic templates
and data files which permitted me to find suitable magic squares. And for the
basic idea to investigate this subject and include me in his endeavor.
Visit his Magic Encyclopedia page on
panmagic stars which deals in more detail with the transformations.
Some definitions All of these but "isolike" and "quadrant magic"
were coined by Aale de Winkel in May of 1999
These magic arrays are discussed in more detail on my Quadrant Magic
Squares page (see below).
panmagic stars 
Order8B star from odd order >5 pandiagonal magic square.
See above secton. 
butterfly star 
An order12 star (points are small) that has 16 or more
lines summing correctly. 
isolike magic star 
Order8B star from odd order >8 quadrant magic square. See
above section. 
incomplete 
A isolike magic star has 10 or 12 lines
of n numbers summing correctly. n is
the order of the magic square. If the 2 main diagonals cannot be made
magic (only 10 correct lines), it is an incomplete isolike magic
star. 
diammagic 
A magic square that has a diamond formation that sums
correctly, in each quadrant. Many of these squares will have diamond
formations in other areas as well! 
plusmagic 
A magic square that has a plus formation that sums
correctly, in each quadrant. Many of these squares will have plus
formations in other areas as well! The order5 pandiagonal magic square
used in my first example has 25 such arrays, one centered around each
number of the square. 
crosmagic 
A magic square that has a plus formation that sums
correctly, in each quadrant. It is not possible to construct panmagic
stars using this formation. 
tcrosmagic 
A form of 'thick' cross that appears in order17 magic
squares. Not usable for panmagic star construction. 
sringmagic 
A ring of n1 numbers around the central number of each
quadrant. These, plus the central number, sum to the constant. It is not
possible to construct panmagic stars using this formation. 
lringmagic 
As sringmagic except there is a gap of 1 number between
each number of the ring. It is not possible to construct panmagic stars
from order9 magic squares using this formation. However, it is possible
with order17 magic squares that have this feature. 
quadrant magic squares 
Magic squares of any order 4n + 1 that contain
any of the above six features (all discovered by Aale de Winkel). While
any given feature must appear in all 4 quadrants, it is quite possible
that this feature will also appear in other areas of the magic square. It
is also possible for a magic square to contain more then one of these
features.
These magic arrays are discussed in more detail on my
Quadrant Magic Squares page. 
Order5 Isomorphic
magic star
Order5 is the only order magic square that can form a
fully isomorphic magic star because there are 25 numbers in the order5
magic square and 25 numbers in a complete order5 isolike magic star.

This star is formed from the magic square below
according to the template below,left. This template can be used to
form such a star from any order5 pandiagonal magic square.
Although this star consists of the consecutive numbers from 1 to 25,
it cannot be considered a normal magic star because it contains five
numbers per line (instead of 4).
Plusmagic
In the course of this investigation, Aale de Winkel noticed
that the magic square shown here has an embedded 'magic' formation.
Five numbers (such as 19, 25, 12, 6 and 3) forming a 'plus' sign sum
to the magic constant.
I subsequently determined that all order5 panmagic squares have this
feature as do many that are not panmagic.
It is the plusmagic feature and not pandiagonal that permits the
transformation of this square to the star. 




1,2 





0,0 

0,1 
0,2 
0,3 

0,4 



1,1 



1,3 



1,0 





1,4 

2,1 
2,0 


2,2 


2,4 
3,3 

3,0 





3,4 



3,1 



3,3 



4,0 

4,) 
4,2 
4,3 

4,4 





3,2 





1 
19 
23 
15 
7 
25 
12 
6 
4 
18 
9 
3 
20 
22 
11 
17 
21 
14 
8 
5 
13 
10 
2 
16 
24 

Regular order  5
This star is constructed from a normal (not pandiagonal),
associative order5 magic square.

9 
2 
25 
18 
11 
3 
21 
19 
12 
10 
22 
20 
13 
6 
4 
16 
14 
7 
5 
23 
15 
8 
1 
24 
17 
This square also is plusmagic, but only in the quadrants. Notice
that the outside diagonals are each formed from one of these 'plus'
arrays. The template above was used for this star also. 
Order7 Panmagic star
The order8 stars shown below are formed from this
order7 pandiagonal magic square. Note the four lighter colored
cells in the two stars below.
This illustrates the problem when we attempt to construct an order7
or higher panmagic star from a pandiagonal magic square.
This is caused because the corner numbers in the square must appear
in 4 different lines; the row, the column, the diagonal and a
pandiagonal.
Example; the 1 appears as part of the diagonal pair 1 and 27 to 31.
Because the outside diagonals are formed from a pandiagonal, these
are panmagic stars.
Because order7 magic squares cannot be plusmagic or diammagic, an
isolike magic star is impossible. 

1 
19 
30 
48 
10 
28 
39 
49 
11 
22 
40 
2 
20 
31 
41 
3 
21 
32 
43 
12 
23 
33 
44 
13 
24 
42 
4 
15 
25 
36 
5 
16 
34 
45 
14 
17 
35 
46 
8 
26 
37 
6 
9 
27 
38 
7 
18 
29 
47 

A. This star is magic in all 12 lines but 4 of the 49 numbers are
not used and 4 numbers are duplicated. 
B. This star uses all 49 numbers (with no duplicates), but only 8
lines sum correctly. The four diamond lines (corners 32, 42, 16 and
13) do not. This is NOT a magic star. 






2,3 











1,4 

1,2 







0,0 

0,5 
0,4 
0,3 
0,2 
0,1 

0,6 





5,5 





5,1 





5,0 



5,3 



5,6 



4,1 
4,0 


4,4 

4,2 


4,6 
4,5 

3,2 

3,0 

3,5 

3,3 

3,1 

3,6 

3,4 

2,1 
2,0 


2,4 

2,2 


2,6 
2,5 



1,0 



1,3 



1,6 





1,5 





1,1 





6,0 

6,5 
6,4 
6,3 
6,2 
6,1 

6,6 







5,4 

5,2 











4,3 







Template notes The numbers in these
templates are coordinates in the relevant magic square.
This template is adapted from Aale de Winkel's isomorphic star
order 8(^{2}_{7}P_{8}^{(2,3)a}) to
create the stars above.
To save space, I have not included the brackets that would normally
appear around each number.

Order9 Isolike magic
star
As explained for the order7, an order9 pandiagonal magic square
cannot form an order8 magic star without duplicating the corner numbers.
However, there is a special kind of order9 pandiagonal that has embedded
formations of 9 cells. Aale de Winkel has labeled them diammagic
(diamond magic) squares. These cells are highlighted in the top left
quadrant of the square used to construct this magic star.

This order8 magic star contains ten lines of nine
numbers.
It uses 65 of the 81 numbers from the magic square and contains no
duplicate numbers.An isolike magic star must also have the two
main diagonal lines also, for a total of 12 correct lines, with no
numbers duplicated. It is impossible to form these diagonal lines
from a diammagic square without requiring duplicate numbers. Iso_9
is, therefore, an incomplete isolike magic star.
14 
40 
64 
30 
56 
8 
79 
27 
51 
9 
78 
23 
49 
10 
39 
65 
35 
61 
44 
70 
36 
60 
5 
76 
19 
48 
11 
75 
20 
53 
16 
45 
69 
32 
58 
1 
67 
28 
57 
2 
80 
25 
54 
15 
41 
24 
50 
13 
37 
66 
29 
62 
7 
81 
34 
63 
6 
77 
22 
46 
12 
38 
71 
47 
17 
43 
72 
33 
59 
4 
73 
21 
55 
3 
74 
26 
52 
18 
42 
68 
31 









2,4 















1,3 

1,5 













3,3 



3,5 









0,0 
0,1 
0,2 

0,3 
0,4 
0,5 

0,6 
0,7 
0,8 






1,0 
1,1 



1,4 



1,7 
1,8 






2,0 









2,8 





2,2 











2,6 



3,1 

3,0 




3,4 




3,8 

3,7 

4,2 


4,0 
4,1 


4,3 
4,4 
4,5 


4,7 
4,8 


4,6 

5,1 

5,0 




5,4 




5,8 

5,7 



6,2 











6,6 





6,0 









6,8 






7,0 
7,1 



7,4 



7,7 
7,8 






8,0 
8,1 
8,2 

8,3 
8,4 
8,5 

8,6 
8,7 
8,8 









5,3 



5,5 













7,3 

7,5 















6,4 










The highlighted cells in the upper left diammagic square above, form
the top left skew line. A similar array in each of the other quadrants
form the other 3 skew lines.
Because the 36 and the 16 also appear in the main diagonal, the two
diagonals cannot be included without requiring duplicate numbers. The
Iso_9 star therefore has only 10 correct lines and cannot be considered
complete.
This same situation occurs with any order 8m + 1 diammagic
square.
The template shown here may be used with any order9 magic square that
is diammagic.
If there are any order9 plusmagic squares, they could be used to form
a complete isolike magic star.

Order9 Butterfly
stars
The overall pattern is an order12 butterfly magic star. It is order12
because of the 12 (small) points.
The first example (butterfly A.) contains 20 correct lines of 9 numbers
with each summing to 369.
It uses all 81 of the original numbers from the square. However, four of
these numbers appear twice.
It must be constructed from a pandiagonal order9 magic square.
Embedded is an order–8 type A star (heavy green lines and slightly
darker cells).
Notice that the line 3 to 25 of the embedded star is composed of the
pandiagonal pair 43  66 and 77 of the magic square.
The 77 already appears in a row, column, and main diagonal, and so must be
duplicated to complete this line.
I have moved the green (duplicate) numbers off of the main diagonals as
shown in de Winkel's template. This results in the main diagonals now also
being correct, but at the cost of a loss in symmetry.
A. 
This is the pandiagonal magic square used for these
butterfly stars. It is diammagic but this feature is not used in this
type of magic star.
2 
43 
78 
67 
21 
35 
54 
59 
10 
60 
13 
3 
44 
81 
68 
19 
29 
52 
30 
53 
63 
14 
1 
38 
79 
69 
22 
72 
23 
28 
47 
61 
15 
4 
39 
80 
37 
74 
70 
24 
31 
48 
62 
18 
5 
16 
6 
40 
75 
71 
27 
32 
46 
56 
49 
57 
17 
9 
41 
73 
65 
25 
33 
26 
36 
50 
55 
11 
7 
42 
76 
66 
77 
64 
20 
34 
51 
58 
12 
8 
45 



1,2 







1,6 



0,0 
0,2 
0,1 
0,3 

0,4 

0,5 
0,7 
0,6 
0,8 

2,1 
2,0 
2,2 

2,3 

2,4 

2,5 

2,6 
2,8 
2,7 

1,0 

1,1 
1.3 

1,4 

1,5 
1,7 

1,8 


3,0 
3,2 
3,1 
3,3 

3,4 

3,5 
3,7 
3,6 
3,8 






8,8 

8,0 






4,0 
4,2 
4,1 
4,3 

4,4 

4,5 
4,7 
4,6 
4,8 






0,8 

0,0 






5,0 
5,2 
5,1 
5,3 

5,4 

5,5 
5,7 
5,6 
5,8 


7,0 

7,1 
7,3 

7,4 

7,5 
7,7 

7,8 

6,1 
6,0 
6,2 

6,3 

6,4 

6,5 

6,6 
6,8 
6,7 

8,0 
8,2 
8,1 
8,3 

8,4 

8,5 
8,7 
8,6 
8,8 



7,2 







7,6 



This template was received from Aale de Winkel May
13/99, his (^{2}_{9}P_{12}^{(1,2)}).
For the Butterfly A. star, the duplicate (green) numbers have been
moved off of the main diagonals. This makes these diagonals sum
correctly, but the star is no longer symmetrical.


Butterfly B. This pattern is an
order12 butterfly magic star quite similar to the above (Butterfly
A.) magic star.
It contains 16 correct lines of 9 numbers with each summing to 369.
It may be constructed from any order9 magic square (not
necessarily pandiagonal).
It is not necessary that the square be quadrant magic.
It uses all 81 of the original numbers from the square.
The same template is used as for the other magic star. The
duplicate (green) numbers have been removed, as have the blank rows &
columns.
There is no longer an embedded order8 magic star. 
Order13 Isolike magic star

This is a pandiagonal diammagic, plusmagic square
of order13. This magic star is constructed from an order13
diammagic square.
It consists of 12 lines of 13 numbers that all sum to 1105.
It uses 107 of the 169 numbers in the magic square, and requires no
duplicate numbers
With 12 correct lines and no duplicate numbers, it is considered
complete.
Only magic squares of orders 8m3 (such as 5, 13, 21, etc)
can form complete isolike magic stars (unless there are
plusmagic squares of order 8m+1).
In this diagram the 3 digit numbers are as large as possible. I
left the 2 digit ones larger for better eligibility. 
155 
117 
66 
28 
159 
121 
83 
45 
7 
138 
100 
62 
24 
165 
127 
89 
51 
13 
131 
93 
55 
17 
148 
110 
72 
34 
6 
137 
99 
61 
23 
154 
116 
78 
27 
158 
120 
82 
44 
16 
147 
109 
71 
33 
164 
126 
88 
50 
12 
143 
92 
54 
39 
157 
119 
81 
43 
5 
136 
98 
60 
22 
153 
115 
77 
49 
11 
142 
104 
53 
15 
146 
108 
70 
32 
163 
125 
87 
59 
21 
152 
114 
76 
38 
169 
118 
80 
42 
4 
135 
97 
69 
31 
162 
124 
86 
48 
10 
141 
103 
65 
14 
145 
107 
79 
41 
3 
134 
96 
58 
20 
151 
113 
75 
37 
168 
130 
102 
64 
26 
144 
106 
68 
30 
161 
123 
85 
47 
9 
140 
112 
74 
36 
167 
129 
91 
40 
2 
133 
95 
57 
19 
150 
122 
84 
46 
8 
139 
101 
63 
25 
156 
105 
67 
29 
160 
132 
94 
56 
18 
149 
111 
73 
35 
166 
128 
90 
52 
1 

It contains magic diamond arrays in each quadrant such
as the one highlighted (centered around the number 71) and is
therefore diammagic.
These four arrays form the four outer diagonal lines of the magic star
above.This same square, however, is also plusmagic.
It contains arrays in each quadrant similar to the one highlighted in
the upper right corner.
A different magic star could be formed from members of these arrays
by substituting the necessary 8 numbers in each of the four outside
diagonals of the above star.
For example, the upper right diagonal could be 126,
88, 50,138, 148, 158, 12, 143,92,
54, 22, 32, 42. The bold numbers are
the same as the original line. 
Finally, this square is sringmagic and
crossmagic, as shown in the lower two quadrants. However, these
features are of no value to the formation of isolike magic stars. The
sringmagic has 2 extra cells on the quadrant diagonal (in addition to the
center cell). The crosmagic array has one one of the lines of the cross
always falling on a main diagonal. This makes it impossible to form the
main diagonals of the star without requiring duplicate numbers.
The four lime colored cells above are the center cells of each array.
The other cells in each pattern are arranged symmetrically around them.
The center row and center column of the quadrant magic square is common
to 2 adjacent quadrants. This means that for most arrays, there are
numbers that are common to two adjacent arrays.
Conditions necessary for transformation of a quadrant magic
square to an isolike magic star.
 One, and only one, number common to the adjacent magic array.
Example in above square are 126 and 114 for the top left diamagic.
 One, and only one, number (the center number of the quadrant) in
each line of the array that falls on a main diagonal of the magic
square.
For sake of completeness, I show the template for this order13
panmagic star.












3,6 























2,5 
2,6 
2,7 





















1,4 

1,6 

1,8 













0,0 
0,1 
0,2 



0,3 
0,4 
0,5 
0,6 
0,7 
0,8 
0,9 



0,10 
0,11 
0,12 






1,0 
1,1 



1,2 







1,10 



1,11 
1,12 






2,0 

2,2 

2,1 









2,11 

2,10 

2,12 









3,3 











3,9 











5,4 













5,8 









4,5 















4,7 







3,0 

















3,12 





4,1 
4,0 






4,4 

4,6 

4,8 






4,12 
4,11 



5,2 

5,0 







5,5 
5,6 
5,7 







5,12 

5,10 

6,3 
6,2 
6,1 
6,0 






6,4 
6,5 
6,6 
6,7 
6,8 






6,12 
6,11 
6,10 
6,9 

7,2 

7,0 







7,5 
7,6 
7,7 







7,12 

7,10 



8,1 
8,0 






8,4 

8,6 

8,8 






8,12 
8,11 





9,0 

















9,12 







8,5 















8,7 









7,4 













7,8 











9,3 











9,9 









10,0 

10,2 

10,1 









10,11 

10,10 

10,12 






11,0 
11,1 



11,2 







11,10 



11,11 
11,12 






12,0 
12,1 
12,2 



12,3 
12,4 
12,5 
12,6 
12,7 
12,8 
12,9 



12,10 
12,11 
12,12 













11,4 

11,6 

11,8 





















10,5 
10,6 
11,8 























9,6 










Comparison of features
Isolike
 Generating magic square is quadrant magic and order 8n
+ 1 or 8n + 3.
 Resulting star is of pattern 8B with 12 lines of
n numbers summing correctly.
 Magic square is plusmagic (required to form main
diagonals).
 If magic square is diamagic, only 10 lines of n
numbers sum correctly and it is called incomplete.
 No numbers are duplicated but not all numbers are
used.
Isomorphic
 Generating magic square is order4, any type, or
order5 plusmagic only.
 Order4; the star will be type 8B normal with
8 lines of 4 numbers (mapping will vary).
 Order5; the star will be type 8B with 12 lines of 5
numbers.
 In both cases, all numbers are used with no
duplicates.
Panmagic
 Generating magic square is pandiagonal odd order
greater then five.
 Resulting star is of pattern 8B with 10 or 12 lines
of m numbers summing correctly.
 Not all numbers are used and there will be either 4
or 8 numbers duplicated.
 A variation is the butterfly, a 12 pointed
star with 20 lines of 9 numbers summing correctly.
 Note that panmagic stars do not require that the
generating square be quadrant magic, only pandiagonal.
Conclusions and Questions
Some regular (not pandiagonal) magic squares of order5 are plusmagic
and can thus form Isomorphic magic stars.
Are there plusmagic
regular magic squares of other orders?
Are all orders 4n
+ 1 that are plusmagic, also crossmagic?
No.
Note that the crossmagic array is of no use in the construction of
panmagic stars because one arm of the cross always falls on the main
diagonal of the square.
Order5 plusmagic squares may be considered to also be diammagic with
sides of length 2. (Order9 diammagic arrays have length 3, and order13
have length 4.)
There seem to be no pandiagonal order9 plusmagic squares.
Are there any regular order9 magic squares
that are plusmagic ?
Are there any plusmagic squares of any order 8m+1?
Yes.
Addendum. June 18,1999.
Other orders 4n + 1 may have either crossmagic or
plusmagic or possibly both in the same square.
There are order17 plusmagic squares which makes the construction of
order17 complete isolike magic stars possible. 
Summary
Isolike magic stars are all of pattern order8B. The
pattern consists of superimposed squares, one of which is rotated 45
degrees. Each of these squares is bisected in both directions by
additional lines.
Construct the normal square from the original top and bottom row, and the
left and right column of the magic square. The four lines of the 45 degree
rotated square is constructed from the diammagic or plusmagic array in
each quadrant of the magic square. Of course the patterns used for a
particular star must all be crossmagic or all diammagic.
The center vertical and horizontal lines are from the corresponding lines
of the magic square.
The two internal diagonal lines are also form the main diagonals of the
magic square.
An incomplete isolike magic star results if the two
main diagonals cannot be made magic without the use of duplicate numbers,
and only 10 lines sum correctly.
The problem arises when more then one number in the plusmagic or diammagic
array is on the main diagonal of the magic square. The magic star then
will require duplicate numbers in order for the main diagonals to be
magic. This happens with the diammagic square of order 8n + 1. It
also happens with all orders of crossmagic and sringmagic squares.
The reason lringmagic squares cannot produce isolike magic stars, is
because more then 1 number of the array is common to the adjacent array.
Order5 pandiagonal (and some regular) magic squares
can form fully isomorphic magic stars.
Orders 8m3 (such as 13, 21, etc) can form
complete isolike magic stars.
Orders 4m1 magic squares cannot form isolike
magic stars because there can be no quadrant magic square for these
orders.
Orders 8m+1 (such as 9, 17, etc) diammagic squares
cannot form complete panmagic stars because some of the numbers on the
side of the diamond falls on the main diagonal of the square, which would
require duplicate numbers in order to form the diagonals. If, however,
there are plusmagic squares of these orders, complete isolike magic stars
could be constructed from them. (See addendum above.)
Isolike magic stars cannot be constructed from the other three types of
quadrant magic squares. crossmagic and sringmagic squares have more then
one number of the array falling on the main diagonal. Lringmagic squares
have more then one number in common with an adjacent array.
Note that panmagic stars can be constructed from pandiagonal magic
squares that are not plusmagic or diammagic. This was de Winkel's original
intention. However, it is impossible to compose 12 line panmagic stars of
this type without using duplicate numbers. This was explained when
describing the order7 situation.
See my page on Quadrant Magic Squares.
Finally, my thanks again to Aale de Winkel for the basic idea, and all
the help I received in developing isolike magic stars.
A final Note:
A fully isomorphic magic star may be constructed from an order 4
magic square.
It is a pure type 8A star containing 8 lines of 4 numbers. See an example
on my Unusual Magic Squares page.
