

I hope you enjoy these examples of a variety of
magic squares.
This large section of my site consists mostly of examples, with a
minimum of explanation and theory.
Two other large sections extend the hypercube discussion to 3 and 4
dimensions
The other main sections deal with magic stars and other number patterns
.
This site should be of interest to middle and high school students and
teachers, and anyone interested in recreational mathematics.
Editor's note 2010: This is the first site I posted to the Internet (in
1998). It
is unorganized because pages were posted as they were written, with no
preplanning.
I have chosen to reproduce them as they were, with almost no editing

July 2011. Two pages on
even order quadrant magic squares 
Order3
Just 1 basic solution. 
2 
7 
11 
14 
16 
9 
5 
4 
13 
12 
8 
1 
3 
6 
10 
15 
Order4
# 290 of 880 basic solutions. 
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 
Order5
# 1233 of 3600 pandiagonal solutions 
Magic Squares are a form of number pattern that has been around for
thousands of years.
For a pure or normal magic square, all rows, columns,
and the two main diagonals must sum to the same value and the numbers used
must be consecutive from 1 to n^{2}, where n is
the order of the square. Many variations exist that contain numerous other
features.
I show on these pages samples of the large variety of magic squares. My
discussions will be limited to brief comments on the individual
illustrations. Perhaps in the future, I will add more in depth information
in the way of history, theory, construction methods, etc.
Acknowledgments: As with all the material on this
site, most of these illustrations are original with myself or I consider
them in the public domain (i.e. I have multiple sources for the
illustration). Many of the more unusual figures are one of a kind and I so
acknowledge the author with thanks for permission to use them. 
Contents

Together use the numbers 1 to 50. 

This and next magic square by
John Hendricks. Order 3 is diamond, 7 & 9 frames. 

This was assembled from
boilerplate sets. Ten different magic squares (in this case). 

An order4 & an order5 combine
to make an order9 magic square. 

This is a simple pure magic
square based on the cyclic number 19. 
Following
are the other magic square pages on this site
1.
A Deluxe Magic Square 
How many groups = 65 in this
Order5 Pandiagonal, Associative, Complete & Selfsimilar Magic Square?
Also, some definitions. 
2.
More Magic Squares 
The Lhoshu, ixohoxi, a 12
squares combination, magic circle, etc. 
3.
More Magic Squares2 
Inlaid, patchwork, gnomon,
topographical, multimagic, Durer, etc. (updated
Mar. 9, 2012) 
4.
Material from REC 
Some magic squares from
Recreational & Educational Computing newsletter. 
5.
Unusual Magic Squares 
A variety of magic squares. A
pandiagonal magic square generator. 
6.
John Hendricks  Cubes 
Some of his large variety of inlaid magic squares, cubes,
and hypercubes. 
7.
Prime Number Magic Squares 
A variety of magic squares constructed with prime numbers. 
8.
Quadrant Magic Squares 
A magic pattern appears in each quadrant. There are many
such patterns. 
a.
Order13 Quadrant Magic S. 
Examples. 
b.
Order17 Quadrant Magic S. 
Examples. 
9. Evenorder
Quadrant Magic Squares 
The even order equivalent of the above quadrant
magic squares 
a.
Order16 Quadrant Magic S. 
All 52 symmetrical patterns and 4 example order16
QMS 
10.
Type2 Order3 Magic Squares 
Turns out the order3 comes in two varieties. i.e. two
different layouts. 
11.
Antimagic Squares 
Examples of different orders of antimagic and
heterosquares. 
12.
Selfsimilar Magic Squares 
Magic squares that produce copies of themselves. 
13.
Mostperfect Magic Squares 
A subset of pandiagonal magic squares that possesses
additional features. 
a.
Mostperfect bentdiagonal 
Franklin type magic squares that are also mostperfect
pandiagonal. 
14.
Magic Square Models 
Photos of models of 3_D magic star, order3 magic cube,
etc. 
15.
Transformations and Patterns 
40+ methods to transform an order4 magic square. Also lists
and groups. 
a.
More order4 Transformations 
Still more transformations. 
b.
Transformations Summary 
A summary of the above. 
c.
R. Fellows Transformations 
Ralph Fellows has found still more transformations. 
16.
Order4 Magic Squares 
Dudeney group patterns. Groups I, II, III, XI and XII in
magic square format. 
a.
Order4 # 1 to 200 
Magic squares in index order, in a tabular list format. 
b.
Order4 # 201 to 400 
Magic squares in index order, in a tabular list format. 
c.
Order4 # 401 to 600 
Magic squares in index order, in a tabular list format. 
d.
Order4 # 601 to 880 
Magic squares in index order, in a tabular list format. 
17.
Pandiagonal Order5 m.s. 
Lists 36 essentially different squares. Each of these has
100 variations. 
18.
Franklin Squares 
3 traditional magic figures plus 3 new, including the
recently discovered 16x16. 
19.
Multimagic Squares 
The new Order12 Trimagic, new tetra and
pentamagic squares, new bimagic cube. 
20.
Perimeter Magic Triangles 
Perimeter magic triangle examples, plus some math. 
21.
Perimeter Magic Polygons <k=3 
Perimeter magic squares, hexagons, and pentagons. 
22.
Magic 3D Polygons and Graphs 
More elaborate magic figures. 
23. Per. Magic Platonic Solids 
Translation of Bao Qishou's 1880 book. 
24.
Magic Knight Tours 
Tracing a path with chess knight moves such that the
numbered steps form a magic square. 
25.
Compact Magic Squares 
Some order8 pandiagonal magic squares that have the compact
feature. 
26.
Ultramagic Squares 
Some unusual magic squares designed by Walter Trump. 
27.
Magic Square Update 
2009. 3 new types of m.s., 1040 order4 ?, How Many ?,
Postage stamp 
28. Magic Square Update2 
2010. A variety of new information and old unique items. 
29. Sparse Magic
Squares 
Some examples of magic squares with some
blank cells. 
Set
of Orders 3, 4, and 5
These three simple magic squares together use the
numbers from 1 to 50.
None of the three is a pure magic square because
none uses consecutive numbers starting at 1.
However, the order 5 square is pandiagonal.
S_{3} = 69, S_{4} = 102, S_{5} = 132 
4 
26 
50 
15 
37 
48 
13 
40 
2 
29 
38 
5 
27 
46 
16 
25 
49 
14 
41 
3 
17 
39 
1 
28 
47 

6 
33 
21 
42 
44 
19 
31 
8 
43 
20 
32 
7 
9 
30 
18 
45 

11 
34 
24 
36 
23 
10 
22 
12 
35 

Orders 3, 5, 7, 9 Inlaid
John R. Hendrick's inlaid magic squares An
order9 magic square with three inlaid magic squares of Orders 3, 5,
and 7. The order3 is rotated 45 degrees and is referred to as a
diamond inlay. Note that the smaller and larger numbers are mixed
throughout the square, not in the outside border as
they would be with a bordered magic square.
These outside rings are called expansion bands to differentiate
them from the borders (of a bordered or concentric magic square),
which have 2n+2 low and high numbers in the border .
S_{3} = 123, S_{5} = 205, S_{7} = 287, S_{9}
= 369.
Numbers used are 1 to 81, so Order9 is a pure magic square.


Order20 with 4 Inlays
I assembled this from a boilerplate design by John
Hendricks. He provides the frame, and four of each of the order7
inlays,
one for each quadrant. It is then simply a matter of deciding which
type of inlay to put in each quadrant.
The order7 (upper right corner) is pandiagonal magic so may be
altered by shifting rows or columns.
The order5 (lower left quadrant) is also pandiagonal magic
The order20, because it contains the consecutive numbers from 1
to 400, is a pure magic square. 
400 
9 
16 
13 
18 
2 
7 
4 
10 
6 
395 
391 
397 
394 
399 
383 
388 
385 
12 
381 
161 
232 
225 
228 
223 
239 
234 
237 
231 
235 
166 
170 
164 
167 
162 
178 
173 
176 
229 
180 
301 
92 
219 
83 
57 
379 
323 
45 
371 
95 
315 
357 
199 
23 
125 
74 
311 
248 
312 
81 
241 
152 
263 
214 
157 
268 
145 
271 
159 
155 
255 
34 
131 
68 
317 
259 
343 
185 
252 
141 
341 
52 
368 
88 
205 
337 
91 
334 
54 
55 
355 
79 
303 
245 
354 
191 
28 
137 
352 
41 
21 
372 
59 
274 
97 
211 
325 
148 
363 
375 
35 
251 
348 
197 
39 
123 
65 
314 
32 
361 
121 
272 
143 
328 
331 
85 
217 
94 
279 
275 
135 
183 
25 
134 
71 
308 
257 
359 
132 
261 
61 
332 
374 
151 
265 
154 
277 
208 
48 
335 
75 
128 
77 
319 
243 
345 
194 
31 
72 
321 
181 
212 
51 
339 
365 
43 
99 
377 
203 
215 
195 
305 
254 
351 
188 
37 
139 
63 
192 
201 
101 
292 
285 
288 
283 
299 
294 
297 
291 
295 
115 
111 
117 
114 
119 
103 
108 
105 
112 
281 
300 
109 
296 
293 
298 
282 
287 
284 
290 
286 
106 
110 
104 
107 
102 
118 
113 
116 
289 
120 
220 
189 
202 
98 
44 
362 
338 
56 
370 
206 
186 
182 
318 
344 
22 
78 
356 
30 
209 
200 
340 
69 
278 
204 
270 
336 
87 
153 
142 
326 
66 
138 
316 
244 
130 
184 
76 
242 
329 
80 
280 
129 
373 
327 
93 
144 
210 
276 
47 
266 
126 
33 
73 
253 
67 
247 
310 
347 
269 
140 
380 
29 
42 
150 
216 
267 
333 
84 
378 
366 
26 
342 
187 
124 
190 
256 
193 
38 
369 
40 
60 
349 
158 
273 
324 
90 
156 
207 
262 
46 
346 
258 
70 
133 
313 
127 
307 
122 
49 
360 
160 
249 
367 
96 
147 
213 
264 
330 
53 
146 
246 
27 
304 
196 
250 
136 
64 
353 
149 
260 
100 
309 
50 
322 
376 
58 
82 
364 
218 
86 
306 
350 
62 
36 
358 
302 
24 
198 
89 
320 
221 
172 
236 
233 
238 
222 
227 
224 
230 
226 
175 
171 
177 
174 
179 
163 
168 
165 
169 
240 
20 
389 
5 
8 
3 
19 
14 
17 
11 
15 
386 
390 
384 
387 
382 
398 
393 
396 
392 
1 

Magic sums are: U. L. 1477, 1055, 633;  U. R. 1337; 
L. L. 1470, 1050;  L. R. 1330, 950, 570
J.R.Hendricks, Magic
square course (selfpublished) pp290294
Four plus five equals nine

Numbers 1 to 25 arranged as an order5 pandiagonal
pure magic square. Numbers 26 to 41 arranged as an embedded
order4 pandiagonal magic square.
Together, they make an order9 magic square. Any one of the rows
and any one of the columns of the order4 is counted twice.
S_{4} = 134, S_{5} = 65, S_{9} = 199
If we use the series from 70 to 110 instead of 1 to 41, the magic
constant of both order4 and order5 is 410 !
As far as I can determine, this type of magic square originated
with Kenneth Kelsey of Great Britain.

Order18 based on 1/19
The number 19 is a cyclic number with a period of 18 before the digits
start to repeat.
The full term decimal expansion of the prime number 19 when multiplied by the
values 1 to 18, may be arranged in a simple magic square of order18, if the
decimal point is ignored. All 18 rows, columns and the two main diagonals sum to
the same value. S = 81. Of course this is not a pure magic square
because a consecutive series of numbers from 1 to n is not used.
Point of interest: 81 is also a cyclic number (of period 9). 1/81 =
.0123456790123456 ... . Only the 8 is missing. Too bad!
1/19 = 
.0 
5 
2 
6 
3 
1 
5 
7 
8 
9 
4 
7 
3 
6 
8 
4 
2 
1 
2/19 = 
.1 
0 
5 
2 
6 
3 
1 
5 
7 
8 
9 
4 
7 
3 
6 
8 
4 
2 
3/19 = 
.1 
5 
7 
8 
9 
4 
7 
3 
6 
8 
4 
2 
1 
0 
5 
2 
6 
3 
4/19 = 
.2 
1 
0 
5 
2 
6 
3 
1 
5 
7 
8 
9 
4 
7 
3 
6 
8 
4 
5/19 = 
.2 
6 
3 
1 
5 
7 
8 
9 
4 
7 
3 
6 
8 
4 
2 
1 
0 
5 
6/19 = 
.3 
1 
5 
7 
8 
9 
4 
7 
3 
6 
8 
4 
2 
1 
0 
5 
2 
6 
7/19 = 
.3 
6 
8 
4 
2 
1 
0 
5 
2 
6 
3 
1 
5 
7 
8 
9 
4 
7 
8/19 = 
.4 
2 
1 
0 
5 
2 
6 
3 
1 
5 
7 
8 
9 
4 
7 
3 
6 
8 
9/19 = 
.4 
7 
3 
6 
8 
4 
2 
1 
0 
5 
2 
6 
3 
1 
5 
7 
8 
9 
10/19= 
.5 
2 
6 
3 
1 
5 
7 
8 
9 
4 
7 
3 
6 
8 
4 
2 
1 
0 
11/19= 
.5 
7 
8 
9 
4 
7 
3 
6 
8 
4 
2 
1 
0 
5 
2 
6 
3 
1 
12/19= 
.6 
3 
1 
5 
7 
8 
9 
4 
7 
3 
6 
8 
4 
2 
1 
0 
5 
2 
13/19= 
.6 
8 
4 
2 
1 
0 
5 
2 
6 
3 
1 
5 
7 
8 
9 
4 
7 
3 
14/19= 
.7 
3 
6 
8 
4 
2 
1 
0 
5 
2 
6 
3 
1 
5 
7 
8 
9 
4 
15/19= 
.7 
8 
9 
4 
7 
3 
6 
8 
4 
2 
1 
0 
5 
2 
6 
3 
1 
5 
16/19= 
.8 
4 
2 
1 
0 
5 
2 
6 
3 
1 
5 
7 
8 
9 
4 
7 
3 
6 
17/19= 
.8 
9 
4 
7 
3 
6 
8 
4 
2 
1 
0 
5 
2 
6 
3 
1 
5 
7 
18/19= 
.9 
4 
7 
3 
6 
8 
4 
2 
1 
0 
5 
2 
6 
3 
1 
5 
7 
8 
This magic square was designed by Harry A. Sayles and published in the
Monist before 1916.
W. S. Andrews, Magic Squares and Cubes,
Dover Publ., 1917, p.176
The next cyclic number (in base 10) that is capable of forming a magic square
in this fashion, is n/383.
In an email dated July 20/01, Simon Whitechapel pointed out that many such
magic squares may be formed using full period cyclic numbers in other bases.
Below we show that the numbers n/19 can be multiplied simply by shifting
left. Obviously, each row and column add to the same value (a property of all
such lists).
1/19 = .0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1
10/19 = .5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0
5/19 = .2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5
12/19 = .6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2
6/19 = .3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6
3/19 = .1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3
11/19 = .5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1
15/19 = .7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5
17/19 = .8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7
18/19 = .9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8
9/19 = 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9
14/19 = .7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4
7/19 = .3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7
13/19 = .6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3
16/19 = .8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6
8/19 = .4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8
4/19 = .2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4
2/19 = .1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2
