# Magic Squares

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.

 8 1 6 3 5 7 4 9 2

Order-3
# 1 of 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 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.

### 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 some of the related pages on this site

### Anti-magic Squares

Examples of different orders of anti-magic and heterosquares.

### Compact magic squares

Some order-8 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 Order-5 Pandiagonal, Associative, Complete & Self-similar 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, order-3 magic cube, etc.

### Material from REC

Some magic squares from Recreational & Educational Computing newsletter.

### More Magic squares-2

A continuation of the above page.

### Most-perfect magic squares

A subset of pandiagonal magic squares that possesses additional features.

### Most-perfect Bent diagonal

Bent-diagonal (Franklin type) magic squares with the added feature that they are most-perfect pandiagonal.

### Multimagic Squares

The new Order-12 Trimagic, new tetra and pentamagic squares, new bimagic cube.

### Order-3 type-2 magic sqrs.

Turns out the order-3 comes in two varieties. i.e. two different layouts.

### Order-4 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.

### Self-similar magic squares

Magic squares that produce copies of themselves.

### The order-5 pandiagonals

Lists 36 essentially different squares. Each of these has 100 variations.

### Transformations & Patterns

40+ methods to transform an order-4 magic square. Also lists and groups.

### Unusual magic squares

A variety of magic squares. A pandiagonal magic square generator.

### Trump - Ultra-magic squares

Some unusual magic squares designed by Walter Trump

### Magic Square Update

3 new types of m.s., 1040 order-4 ?, 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 3-D hypercubes

### Magic tesseracts

11 pages dealing with these 4-D hypercubes (New! November 2007)

### Links to similar web sites

Other Magic Square pages or Recreational Mathematics sites (by other authors).

### Set of Orders 3, 4, and 5

 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

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

### 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 diferentiate 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

 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 (self-published) pp290-294
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 a pandiagonal so may be altered by shifting rows or columns.
The order-5 (lower left quadrant) is also a pandiagonal.
The order-20, 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

### 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 = 0.1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 3/19 = 0.1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 4/19 = 0.2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 5/19 = 0.2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 6/19 = 0.3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 7/19 = 0.3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 8/19 = 0.4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 9/19 = 0.4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 10/19= 0.5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 11/19= 0.5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 12/19= 0.6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 13/19= 0.6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 14/19= 0.7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 15/19= 0.7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 16/19= 0.8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 17/19= 0.8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 18/19= 0.9 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```
`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.`

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Harvey Heinz   harveyheinz@shaw.ca
This page last updated October 05, 2009
Copyright © 1998,1999, 2000 by Harvey D. Heinz