I hope you enjoy these examples of a variety of magic squares.
This section of my site consists mostly of examples, with a minimum of explanation and
theory.
Refer to the other two sections for Magic Stars and Number Patterns.
This site should be of interest to middle and high school students and teachers, and
anyone interested in recreational mathematics.
Order3 
Order4 
Order5 
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 descussions 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 mateial 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.
Set of Orders 3, 4, and 5 
Together use the numbers 1 to 50. 
Orders 3, 5, 7, 9 Inlaid 
This and next magic square by John Hendricks. Order 3 is diamond, 7 & 9 frames. 
Order20 with 4 Inlays 
This was assembled from boilerplate sets. Ten different magic squares (in this case). 
Four plus five equals nine 
An order4 & an order5 combine to make an order9 magic square. 
Order18 based on 1/19 
This is a simple pure magic square based on the cyclic number 19. 
Following 
are some of the related pages on this site 
Antimagic Squares 
Examples of different orders of antimagic and heterosquares. 
Compact magic squares 
Some order8 pandiagonal magic squares that have the compact feature. 
FranklinSquares 
The 3 traditional magic figures plus 3 new, including the recently discovered 16x16. 
How many groups = 65? 
in this Order5 Pandiagonal, Associative, Complete & Selfsimilar Magic Square? 
John Hendricks  Cubes 
Some of his large variety of inlaid magic squares, cubes, and hypercubes. 
Knight tours 
Tracing a path with chess knight moves such that the numbered steps form a magic square. 
Magic Square models 
Photos of models of 3_D magic star, order3 magic cube, etc. 
Material from REC 
Some magic squares from Recreational & Educational Computing newsletter. 
More Magic Squares 
A continuation of this page. 
More Magic squares2 
A continuation of the above page. 
Mostperfect magic squares 
A subset of pandiagonal magic squares that possesses additional features. 
Mostperfect Bent diagonal 
Bentdiagonal (Franklin type) magic squares with the added feature that they are mostperfect pandiagonal. 
Multimagic Squares 
The new Order12 Trimagic, new tetra and pentamagic squares, new bimagic cube. 
Order3 type2 magic sqrs. 
Turns out the order3 comes in two varieties. i.e. two different layouts. 
Order4 Magic Squares 
Dudeney group patterns. Groups I, II, III, XI and XII in magic square format.. All 880 magic squares in index order, in a tabular list format. 
Perimeter magic polygons 
Perimeter magic triangles and other polygons. Plus two subsidiary pages. 
Prime Magic Squares 
A variety of magic squares constructed with prime numbers. 
Quadrant Magic Squares 
A magic pattern appears in each quadrant. There are many such patterns. 
Selfsimilar magic squares 
Magic squares that produce copies of themselves. 
The order5 pandiagonals 
Lists 36 essentially different squares. Each of these has 100 variations. 
Transformations & Patterns 
40+ methods to transform an order4 magic square. Also lists and groups. 
Unusual magic squares 
A variety of magic squares. A pandiagonal magic square generator. 
Trump  Ultramagic squares 
Some unusual magic squares designed by Walter Trump 
Magic Square Update 
3 new types of m.s., 1040 order4 ?, How Many ?, Postage stamp 
Site Map 
Titles and relationship of all pages on this site (and related sites below) with direct links. 
Following 
are related sites (by this author) on other servers 
Magic cubes 
About 45 pages dealing with these 3D hypercubes 
Magic tesseracts 
11 pages dealing with these 4D hypercubes (New! November 2007) 
Links to similar web sites 
Other Magic Square pages or Recreational Mathematics sites (by other authors). 



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
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 diferentiate 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.
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 
J.R.Hendricks, Magic square course (selfpublished) pp290294
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 a pandiagonal so may be altered by shifting rows or
columns.
The order5 (lower left quadrant) is also a pandiagonal.
The order20, because it contains the consecutive numbers from 1 to 400, is a pure magic
square
Magic sums are: U.L. 1477, 1055, 633;  U.R. 1337;  L.L. 1470, 1050;  L. R.
1330, 950, 570
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.
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
Use the links at the top of this page to access other magic square pages on this site.
Thanks again for the visit and I hope to see you again soon.
Please send me Feedback about my Web site!
Harvey Heinz harveyheinz@shaw.ca
This page last updated
October 05, 2009
Copyright © 1998,1999, 2000 by Harvey D. Heinz