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41 pages

B. Magic Cubes
45 pages

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13 pages

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26 pages

E. Number Patterns
5 pages

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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

 

 
 
8 1 6
3 5 7
4 9 2

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

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.

Order-20 with 4 Inlays

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

Four plus five equals nine

An order-4 & an order-5 combine to make an order-9 magic square.

Order-18 based on 1/19

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

This page was originally posted May 1998
It was last updated March 09, 2012
Harvey Heinz   harveyheinz@shaw.ca
Copyright © 1998-2009 by Harvey D. Heinz