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|

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
pre-planning.
I have chosen to reproduce them as they were, with almost no editing
 |
July 2011. Two pages on
even order quadrant magic squares |

Order-3
Just 1 basic solution. |
2 |
7 |
11 |
14 |
16 |
9 |
5 |
4 |
13 |
12 |
8 |
1 |
3 |
6 |
10 |
15 |
Order-4
# 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 |
Order-5
# 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 n2, 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 order-4 & an order-5 combine
to make an order-9 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
Order-5 Pandiagonal, Associative, Complete & Self-similar Magic Square?
Also, some definitions. |
2.
More Magic Squares |
The Lho-shu, ixohoxi, a 12
squares combination, magic circle, etc. |
3.
More Magic Squares-2 |
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.
Order-13 Quadrant Magic S. |
Examples. |
b.
Order-17 Quadrant Magic S. |
Examples. |
9. Even-order
Quadrant Magic Squares |
The even order equivalent of the above quadrant
magic squares |
a.
Order-16 Quadrant Magic S. |
All 52 symmetrical patterns and 4 example order-16
QMS |
10.
Type-2 Order-3 Magic Squares |
Turns out the order-3 comes in two varieties. i.e. two
different layouts. |
11.
Anti-magic Squares |
Examples of different orders of anti-magic and
heterosquares. |
12.
Self-similar Magic Squares |
Magic squares that produce copies of themselves. |
13.
Most-perfect Magic Squares |
A subset of pandiagonal magic squares that possesses
additional features. |
a.
Most-perfect bent-diagonal |
Franklin type magic squares that are also most-perfect
pandiagonal. |
14.
Magic Square Models |
Photos of models of 3_D magic star, order-3 magic cube,
etc. |
15.
Transformations and Patterns |
40+ methods to transform an order-4 magic square. Also lists
and groups. |
a.
More order-4 Transformations |
Still more transformations. |
b.
Transformations Summary |
A summary of the above. |
c.
R. Fellows Transformations |
Ralph Fellows has found still more transformations. |
16.
Order-4 Magic Squares |
Dudeney group patterns. Groups I, II, III, XI and XII in
magic square format. |
a.
Order-4 # 1 to 200 |
Magic squares in index order, in a tabular list format. |
b.
Order-4 # 201 to 400 |
Magic squares in index order, in a tabular list format. |
c.
Order-4 # 401 to 600 |
Magic squares in index order, in a tabular list format. |
d.
Order-4 # 601 to 880 |
Magic squares in index order, in a tabular list format. |
17.
Pandiagonal Order-5 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 Order-12 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 3-D Polygons and Graphs |
More elaborate magic figures. |
23. Per. Magic Platonic Solids |
Translation of Bao Qi-shou'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 order-8 pandiagonal magic squares that have the compact
feature. |
26.
Ultra-magic Squares |
Some unusual magic squares designed by Walter Trump. |
27.
Magic Square Update |
2009. 3 new types of m.s., 1040 order-4 ?, How Many ?,
Postage stamp |
28. Magic Square Update-2 |
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.
S3 = 69, S4 = 102, S5 = 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
order-9 magic square with three inlaid magic squares of Orders 3, 5,
and 7. The order-3 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 .
S3 = 123, S5 = 205, S7 = 287, S9
= 369.
Numbers used are 1 to 81, so Order-9 is a pure magic square.
|

|

Order-20 with 4 Inlays
I assembled this from a boilerplate design by John
Hendricks. He provides the frame, and four of each of the order-7
inlays,
one for each quadrant. It is then simply a matter of deciding which
type of inlay to put in each quadrant.
The order-7 (upper right corner) is pandiagonal magic so may be
altered by shifting rows or columns.
The order-5 (lower left quadrant) is also pandiagonal magic
The order-20, 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 (self-published) pp290-294
 
Four plus five equals nine
 |
Numbers 1 to 25 arranged as an order-5 pandiagonal
pure magic square. Numbers 26 to 41 arranged as an embedded
order-4 pandiagonal magic square.
Together, they make an order-9 magic square. Any one of the rows
and any one of the columns of the order-4 is counted twice.
S4 = 134, S5 = 65, S9 = 199
If we use the series from 70 to 110 instead of 1 to 41, the magic
constant of both order-4 and order-5 is 410 !
As far as I can determine, this type of magic square originated
with Kenneth Kelsey of Great Britain.
|

Order-18 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 order-18, 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 e-mail 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
|