CONTENTS

A brief introduction to
this subject and quadrant magic arrays. 

Examples of magic
quadrant arrays and order5 quadrant magic squares. 

An example shows the
relationship to quadrantmagic squares. 

Examples of magic
quadrant arrays and order9 quadrant magic squares. 

A page of magic quadrant
arrays and order13 quadrant magic squares. 

A page of magic quadrant
arrays and order17 quadrant magic squares. 

Comments, questions, and
a credit (and link) to Aale de Winkel. 

A new
page discusses even order QMS, appearing 12 years later 

3
additional example squares demonstrating newly discovered facts by
Dwane Campbell 
Introduction
Some magic squares of orders m equal to 4n + 1, have
arrays of m cells appearing in each quadrant that sum to the magic
constant.
If a magic square contains 4 of these arrays in the 4 quadrants, and if
they are all the same type, I call it a quadrantmagic square.
Consider a magic square of order m = 4n + 1 (i.e. 5, 9, 13,
17, etc).
Divide it into 4 quadrants such that each quadrant consists of (m + 1)/2
times (m + 1)/2 cells
Each array consists of m 1 cells, plus the central cell of the quadrant.
The array is considered magic if the m cells sum to the magic constant
for the square.
Quadrant Magic Arrays
The cells in the array must be arranged so that they are orthogonally and
diagonally symmetrical. This condition reduces the number of possible magic
arrays to a manageable number (at least for the smaller magic squares).
Note that the central row and the central column of the Quadrant Magic Square is
common to two orthogonally adjacent quadrants. This means that if an array has
cells in the outside row and column, these cells are shared with the adjacent
quadrant.
The first 6 magic arrays were all discovered by Aale de Winkel in May, 1999.
They are the cross, plus sign, diamond, small ring, large ring and thickcross.
He named them respectively, crosmagic, plusmagic diammagic, sringmagic,
lringmagic and tcrosmagic.
After I found some additional arrays, he decided to investigate the subject
systematically.
It turns out there are 10 classes of arrays, determined by their degree of
symmetry. Because of the very large number of magic arrays, we define a
Quadrant Magic Square as using only the highest order one of these,
the fully symmetric one. This class we call 'Quadrant' or
quadrant magic array.
Aale enumerated the Quadrant magic patterns for orders 5, 9, 13 and 17 and
labeled them with index numbers prefixed with a 'p'. For order 13 there are 38
quadrant magic patterns, but a total of 262,596,783,764 patterns counting all 10
classes of magic arrays.
See his Special_Magic.html page for details and listings. (The link to his site
appears at the bottom of this page.)
The first five of the above named arrays are fundamental.
They appear in all orders 4n + 1 (altered, of course, to contain the
correct number of cells).
Because the 'p' numbers for these patterns vary from order to order, these names
will be retained.
Here are the fundamental arrays for order9, with the corresponding 'p' number.
plusmagic (p1)

sringmagic (p2)

diammagic (p3)

lringmagic (p5)

crosmagic (p6)

Quadrant Magic Squares
Some quadrant magic squares may be converted to isomorphiclike magic
stars. For order5, they are isomorphic. For the other orders, they are
only pseudoisomorphic because they cannot use all the numbers contained in the
quadrant magic square.
The only magic array that may be used to form an isolike
magic star for all orders 4n+1 is the plusmagic.
Additionally, the diammagic array can form isolike
magic stars for orders 8n – 3.
The reasons for this are discussed in the isolike magic star section.
On this page, I will show examples of quadrantmagic squares, present known
and suspected characteristics, and pose a number of questions for further study.
I will also present an example of the order5 isomorphic magic star to show how
the quadrant magic square arrays are used to create these stars.
As my first example, here is an order17
pandiagonal magic square that I show as quadrant magic four different ways by
illustrating a different array in each quadrant. However, each array actually
appears in all four quadrants.
As well as the 17 rows and columns, and the 34 diagonals, each of these arrays
sum to the magic constant of 2465.
271 
285 
10 
24 
38 
52 
83 
97 
111 
125 
139 
170 
184 
198 
212 
226 
240 
221 
235 
249 
263 
277 
2 
33 
47 
61 
75 
89 
103 
134 
148 
162 
176 
190 
154 
185 
199 
213 
227 
241 
272 
286 
11 
25 
39 
53 
84 
98 
112 
126 
140 
104 
135 
149 
163 
177 
191 
205 
236 
250 
264 
278 
3 
34 
48 
62 
76 
90 
54 
85 
99 
113 
127 
141 
155 
186 
200 
214 
228 
242 
256 
287 
12 
26 
40 
4 
18 
49 
63 
77 
91 
105 
136 
150 
164 
178 
192 
206 
237 
251 
265 
279 
243 
257 
288 
13 
27 
41 
55 
69 
100 
114 
128 
142 
156 
187 
201 
215 
229 
193 
207 
238 
252 
266 
280 
5 
19 
50 
64 
78 
92 
106 
120 
151 
165 
179 
143 
157 
171 
202 
216 
230 
244 
258 
289 
14 
28 
42 
56 
70 
101 
115 
129 
93 
107 
121 
152 
166 
180 
194 
208 
222 
253 
267 
281 
6 
20 
51 
65 
79 
43 
57 
71 
102 
116 
130 
144 
158 
172 
203 
217 
231 
245 
259 
273 
15 
29 
282 
7 
21 
35 
66 
80 
94 
108 
122 
153 
167 
181 
195 
209 
223 
254 
268 
232 
246 
260 
274 
16 
30 
44 
58 
72 
86 
117 
131 
145 
159 
173 
204 
218 
182 
196 
210 
224 
255 
269 
283 
8 
22 
36 
67 
81 
95 
109 
123 
137 
168 
132 
146 
160 
174 
188 
219 
233 
247 
261 
275 
17 
31 
45 
59 
73 
87 
118 
82 
96 
110 
124 
138 
169 
183 
197 
211 
225 
239 
270 
284 
9 
23 
37 
68 
32 
46 
60 
74 
88 
119 
133 
147 
161 
175 
189 
220 
234 
248 
262 
276 
1 

The arrays are:
sringmagic (p082) plusmagic (p001)
p085 lringmagic (p213)
not shown is the p216 (and surely many others)
The center numbers in each quadrant are:
127 256
16 145
The arrays are centered around them and they are one of the
m cells of the array.Because numbers 200 and
72 are on the center column, they are common to 2
adjacent horizontal plusmagic arrays.
(Of course, these same numbers are common to 2 horizontally adjacent
lringmagic arrays also.)
Numbers 216 and 56 which on the center
row are each common to two vertically adjacent plusmagic arrays. 
With the lringmagic arrays, there are five numbers common to each of two
orthogonal arrays.
This is LP (14, 1, 0)(1, 14, 0).
I will discuss the characteristics of these arrays and quadrant magic squares
in more detail as I introduce the different orders.
Suffice to say that in order to qualify as a quadrant magic square:
 The square must be magic in the ordinary sense i.e. all rows, columns and
the two main diagonals must be magic.
 The magic square may be of any type i.e. normal, pandiagonal, associated,
inlaid, etc.
 All four quadrants of the square must contain the same magic array of
m numbers, and it must be centered around the central number of the
quadrant.
 The array mentioned in statement 3 must be quadrant magic i.e. it must be
fully symmetrical.
 Statement three requires that the magic square be of order 4m + 1.
Notes:
 Most magic squares will contain one (or more) magic arrays, but in only 1,
2, or 3 quadrants (or an array not centered in a quadrant). Also, many magic
squares will contain a pattern in all 4 quadrants that is not fully
symmetrical.
These squares are not quadrant magic squares!
 In each order, the middle row of the magic square is common to both the
top 2 quadrants and the bottom 2 quadrants. Likewise, the middle column is
common to the pairs of quadrants on the left and right sides.
Order5
quadrant magic squares
Quadrant diagrams
Order5 essentially different pandiagonal ................... crosmagic,
plusmagic
Order5 1 of the 99 derivatives of above .................. crosmagic,
plusmagic
Order5 normal (not pandiagonal) ............................ no quadrant
magic arrays
Order5 normal (not pandiagonal) ............................ crosmagic,
plusmagic
Order5 normal associative ...................................... only 2
quadrant magic arrays
Order5 normal associative ......................................
plusmagic, & 2 crosmagic
Order5 pandiagonal associative .............................. plusmagic,
& 2 crosmagic
plusmagic

For order5 each
quadrant is 3 by 3 cells. Because of the
small number of cells in the order5 quadrant, there are only 5 quadrant
magic arrays possible: The plusmagic and diammagic (which are the same for
this order), and the crosmagic, sringmagic and lringmagic (which are the
same for this order).
All 36 essentially different order5 pandiagonal magic
squares are plusmagic. In fact, there is a magic array of 5 cells centered
around each of the 25 cells of each of these magic squares (using
wraparound when necessary).
From the above I think we can safely assume that all
3600 order5 pandiagonal magic squares are plusmagic. However, all order5
plusmagic are NOT pandiagonal (see examples 4 & 6 below). 
crosmagic

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

This is essentially different pandiagonal magic square #
10 (of 36). This square is plusmagic. In fact, there
is a magic array centered on each of the 25 cells of the magic square. It
may also be considered diammagic (the diamonds have sides of length 2).
Notice that the 21, 5, 13 and 17 are each shared by two arrays. This is
important in the construction of isolike magic stars.
It is also crosmagic and may also be considered sringmagic
and lringmagic, which is the same configuration for order5. 
3 
7 
14 
16 
25 
11 
20 
23 
2 
9 
22 
4 
6 
15 
18 
10 
13 
17 
24 
1 
19 
21 
5 
8 
12 

This is one of the 99 pandiagonal derivations of the
above square. It was obtained by transformation
13524 applied to the rows and columns: then rows and columns
interchanged with the diagonals; then another transformation 13524
applied to the rows and columns.
See Benson & Jacoby, New Recreations With Magic Squares, Dover, 1976, p.130.
This one shows crosmagic arrays. However, both this and
the previous magic square contain four of each of these two magic arrays. 
17 
13 
5 
6 
24 
9 
25 
11 
2 
18 
21 
4 
8 
20 
12 
3 
16 
22 
14 
10 
15 
7 
19 
23 
1 

This is an normal (not
pandiagonal) magic square. It is not quadrant
magic. There are no magic arrays. 
1 
7 
19 
13 
25 
18 
15 
21 
2 
9 
22 
4 
8 
20 
11 
10 
16 
12 
24 
3 
14 
23 
5 
6 
17 

This is a normal (not
pandiagonal) magic square. This is a plusmagic
quadrant magic square.
Normal magic squares that are quadrant magic seem to be
relatively rare. 
12 
1 
20 
9 
23 
21 
15 
4 
18 
7 
10 
24 
13 
2 
16 
19 
8 
22 
11 
5 
3 
17 
6 
25 
14 

This is a normal (not
pandiagonal) associated magic square It is not
quadrant magic because only two quadrants have a plusmagic array.
It seems that there is always zero, two or four
quadrants correct in order5 magic squares. 
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 is a normal (not
pandiagonal) associated magic square This is a
plusmagic quadrant magic square. There are two crosmagic arrays, but only
two (see them?), so this square is not a crosmagic quadrant magic square. 
1 
15 
24 
8 
17 
23 
7 
16 
5 
14 
20 
4 
13 
22 
6 
12 
21 
10 
19 
3 
9 
18 
2 
11 
25 

This is a pandiagonal
associated crosmagic square The crosmagic array
cannot be used to form an order5 isomorphic star for 2 reasons
 There are two cells in common with the adjacent array
instead of one.
 There are three cells on the diagonal instead of one.
This square also is plusmagic (and diammagic), so an
order5 isolike magic star may be made using these arrays. 
Order5 ...some conclusions and questions
Order5 has only five magic arrays because of the small size of the
quadrants, and only two of these are unique.
The plusmagic and diammagic arrays are identical, as are the crosmagic,
sringmagic and lringmagic arrays.
All 36 essentially different pandiagonal magic squares are plusmagic on
all 25 cells.
Does this apply to the 99 variations of each of
these?
Only some regular order5 are plusmagic.
Are any of these plusmagic for all 25 cells?
Are all of the nine pandiagonal associated magic
squares plusmagic?
Are any of the regular associated magic squares
plusmagic?
Isolike magic Stars
A. Plusmagic
18 
21 
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7 
15 
2 
10 
13 
16 
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19 
22 
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1 

B. Crosmagic
18 
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15 
2 
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16 
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19 
22 
5 
8 
25 
3 
6 
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17 
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1 

The magic star shown below
is isomorphic to magic square A.
Each number in the magic square is mapped to a location in the star.
Order5 is the only size of quadrant magic square that can be
transformed to a magic star using all the numbers contained in the
square. For that reason, I use the general term 'Isolike magic
stars' to cover all orders. 

Orders 9, 13, 17, etc magic squares may be used to form
this type of star and only if the square is quadrantmagic. Only 25 of the
numbers in the magic square can be used, however. Finally, for a given
order, only certain magic arrays can be used to form such a star. This
order8 type B star has 12 lines of 5 numbers summing to the magic
constant, the same as the order5 magic square.
The outside horizontal and vertical lines (I call these the ‘square’)
contain the same numbers in the same order as the outside rows and columns
of the magic square.
This predetermines two of the numbers, such as the 2 and 21, in each
outside diagonal line (I call this the diamond).
The corner numbers of the diamond are those that are common to two
magic arrays in the quadrant magic square. In this case the 13, 19, 5 and
6 in the plusmagic arrays of square A. This leaves just one number in each
side of the diamond to be assigned, and this is the fifth number of the
corresponding magic array. 
Square B shows the crosmagic arrays. On close examination we
see two reasons why an isolike magic star cannot be formed from this array.
 There are 2 numbers in each array common to the adjacent
array instead of 1.
 There are 3 numbers in each diagonal of the array instead
of 1.
For any order quadrant magic square the plusmagic
array may be used to convert the square to an isolike magic star.
For orders 8m  3 the diammagic array may be used but
orders 8m+1 fail because of the additional diagonal cells.
All other quadrant magic arrays fail, for
every order, due to either or both situations mentioned above.
Please see my Isolike Magic Stars
page for more details and examples of orders 5, 9 and 13 stars.
Order9
quadrant magic squares
Starting with this order, we identify the patterns by
their index numbers.
p1 (plusmagic)

Quadrant magic arrays
The same array must appear in all four quadrants of the magic square
for it to be called a Quadrant magic square!Quadrant
magic squares using this array can form isolike magic stars.
However, I have not found such magic squares in order9. 
p2 (sringmagic)

A quadrant magic square with this array cannot be
transformed into an isolike magic star because of the 3 cells
(instead of 1) that appear on the diagonals. So far, all Quadrant
Magic squares found using this array are also lring quadrant magic. 
p3 (diammagic)

This array also cannot be used to form an isolike
magic star because of the 3 cells (instead of 1) that appear on the
diagonals.
There are diammagic quadrant magic squares. 
p4

This array cannot be used to form an isolike magic star due to both
of the reasons explained above.So far, I have found no Quadrant
Magic squares using this array.
Aale de Winkel found this pattern on Aug. 31, 1999. He also
showed mathematically that there can be only 7 totally symmetric
patterns for order9. 
p5 (lringmagic)

This array cannot be used to form an isolike magic
star because of the 3 cells (instead of 1) that appear on the
diagonals. Also, 3 cells instead of 1 appear in the outside rows and
columns, and so are common to orthogonally adjacent quadrants. 
p6 (crosmagic)

This array cannot be used to form an isolike magic
star because of the 5 cells (instead of 1) that appear on the
diagonals. Also, 2 cells instead of 1 appear in the outside rows and
columns, and so are common to orthogonally adjacent quadrants. In
any case, I have not found such magic squares in order9. 
p7

This array cannot be used to form an isolike magic
star because 2 cells instead of 1 appear in the outside rows and
columns, and so are common to orthogonally adjacent quadrants. So
far, I have found no Quadrant Magic squares using this array. 
A pandiagonal sring, lring quadrant magic square
55 
25 
40 
62 
23 
38 
60 
21 
45 
69 
3 
54 
64 
7 
49 
71 
5 
47 
17 
32 
74 
15 
30 
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10 
34 
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19 
43 
58 
26 
41 
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24 
39 
63 
6 
48 
72 
1 
52 
67 
8 
50 
65 
35 
77 
11 
33 
75 
18 
28 
79 
13 
37 
61 
22 
44 
59 
20 
42 
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27 
51 
66 
9 
46 
70 
4 
53 
68 
2 
80 
14 
29 
78 
12 
36 
73 
16 
31 

Here the lime green cells are the center of each quadrant
and are 1 of the 9 cells in each array.
The yellow cells are the lringmagic arrays. I show only 2 of the 4
quadrants for clarity (in the upper left and lower right quadrants).
The blue cells are the sringmagic arrays.An order9 sringmagic square
cannot form an isolike magic star because 3 cells instead of 1 are on the
main diagonal.
An order9 lringmagic square cannot form an isolike magic star for the
same reason given above. Also 3 cells instead of 1 are common with the
adjacent quadrant.
All ringmagic quadrant magic squares found to date are
both sringmagic and lringmagic (both p2 and p5).
(Until July 2011.) 
A pandiagonal diammagic quadrant magic square
45 
79 
71 
20 
30 
46 
58 
14 
6 
35 
47 
57 
10 
4 
41 
78 
72 
25 
3 
37 
76 
68 
24 
36 
52 
62 
11 
22 
32 
51 
63 
16 
8 
38 
75 
64 
15 
9 
43 
80 
65 
21 
28 
49 
59 
70 
26 
29 
48 
55 
13 
5 
42 
81 
56 
12 
1’ 
40 
77 
69 
27 
34 
53 
73 
67 
23 
33 
54 
61 
17 
2 
39 
50 
60 
18 
7 
44 
74 
66 
19 
31 

An order9 diammagic square cannot form an isolike magic
star because 3 cells instead of 1 are on the main diagonal. These two
types of order9 quadrant magic squares are the only ones found to date. 
Order9 quadrant magic square
questions
There are 7 quadrant magic arrays (totally symmetric patterns)
for order9.
So far only 2 types of quadrant magic squares have been found.
Do sringmagic (P2)
and lringmagic (P5) arrays
always appear together?
No. See
addendum example square 2 and 3.
Do diammagic (P3)
arrays never appear with
sringmagic (P2) or lringmagic (P5)
squares?
No. See
addendum example square 1.
Are there NO crosmagic
(P6), plusmagic (P1), P4
or P7 order9 quadrant magic squares?
No for P4. See
addendum example square 1.
Are all order9 quadrant magic squares
pandiagonal?
Quadrant Magic
Squares Summary
Orders 13 and 17 are on separate pages due to amount of
material. See order13 and
order17
My search for quadrant magic squares was performed mostly
using Latin prescriptions.
It would be interesting to see results of searches using other methods of magic
square generation.
Most of the quadrant magic squares found (and all orders 9 and 13) are
pandiagonal .
Are many regular magic squares quadrant magic?
Are many associated magic squares quadrant magic?
Can there be quadrant magic squares of even order?
(the quadrants would be n/2 with no central cell in the array.)
All quadrant magic squares found to
date have been a result of searching.
Can an algorithm be developed to generate
quadrant magic squares?
Order 
Number of magic arrays on these
pages 
Quadrant magic 
Order13 seems to have the most densely packed
quadrant magic squares.
Order13 has a 14way quadrant magic square. The best I can find for
order17, which should have a great many more combinations, is a 6way
quadrant magic square.So far, all order13 Quadrant m. s. found are
pandiagonal, although regular m. s. have been found for orders 5 and 17. 
5 
5 but only 2 are unique 
2 way quadrant magic 
9 
7 
2 way quadrant magic 
13 
38 
14 way quadrant magic 
17 
15 ( of a total of 253) 
6 way quadrant magic 
Credit
I wish to thank Aale de Winkel for discovering these
fascinating magic squares and for all the help he has given me in my
attempts to consolidate the features of quadrant magic squares and isolike
magic stars.
As well as suggestions, he provided me lists of his Latin prescription
squares where he searched for these features, and a program to convert any
Latin prescription (LP) to an actual magic square.
Please visit his site at
http://www.magichypercubes.com/Encyclopedia/index.html then link to
quadrantmagic and specialmagic.
Addendum  July 2011
In July 2011, as a result of
working on evenorder quadrant magic squares, Dwane Campbell decided to look at
some of the questions proposed on this page.
The following three order9 quadrant magic squares provide answers for 3 of the
4 questions I proposed for order9.
Example 1
1 
14 
72 
64 
77 
54 
46 
32 
9 
38 
61 
24 
20 
43 
60 
56 
25 
42 
80 
48 
31 
35 
3 
13 
17 
66 
76 
8 
12 
67 
71 
75 
49 
53 
30 
4 
45 
59 
19 
27 
41 
55 
63 
23 
37 
78 
52 
29 
33 
7 
11 
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70 
74 
6 
16 
65 
69 
79 
47 
51 
34 
2 
40 
57 
26 
22 
39 
62 
58 
21 
44 
73 
50 
36 
28 
5 
18 
10 
68 
81 
This square is quadrant magic with P2, P3, P4 and P5
in all 4 quadrants. I show one of each here.
No other patterns appear in the square.
It is also associated, {compact(solid_3x3)}, and
complete_3. 
Example 2
1 
71 
46 
50 
3 
68 
72 
49 
9 
51 
26 
60 
70 
39 
25 
2 
58 
38 
63 
37 
14 
19 
59 
36 
41 
27 
73 
17 
78 
35 
30 
16 
75 
76 
29 
13 
28 
6 
64 
77 
52 
5 
18 
65 
54 
67 
53 
21 
8 
66 
40 
48 
4 
62 
24 
55 
42 
43 
23 
61 
56 
45 
20 
44 
10 
80 
57 
32 
12 
22 
81 
31 
74 
33 
7 
15 
79 
47 
34 
11 
69 
This example shows the P2 pattern, which appears in
all 4 quadrants.
No other patterns appear in any quadrant.

Example 3
1 
9 
4 
51 
47 
54 
71 
67 
65 
42 
38 
43 
61 
58 
57 
19 
27 
23 
81 
76 
75 
11 
18 
14 
31 
29 
34 
2 
7 
5 
49 
48 
52 
72 
68 
66 
40 
39 
44 
63 
59 
55 
20 
25 
24 
79 
77 
73 
12 
16 
15 
32 
30 
35 
3 
8 
6 
50 
46 
53 
70 
69 
64 
41 
37 
45 
61 
60 
56 
21 
26 
22 
80 
78 
74 
10 
17 
13 
33 
28 
36 
This example shows the P5 pattern, which appears in
all 4 quadrants.
The P2 pattern appears in only 2 quadrants.
No other patterns appear in the square.

Three questions asked in July
1999. (Repeated here for convenience.)
Do sringmagic (P2)
and lringmagic (P5) arrays
always appear together?
No. See example square 2 and 3.
Do diammagic (P3)
arrays never appear with
sringmagic (P2) or lringmagic (P5)
squares?
No. See
example square 1.
Are there NO crosmagic
(P6), plusmagic (P1), P4
or P7 order9 quadrant magic squares?
No for P4. See
example square 1.
