MAGIC SQUARES
Lies De Sutter
An De Brandt
Katrien De Bruycker
Vicky De Marteau
Why this subject ?
When we think about Mathematics we think most of the time of long lessons but there is more…
One day our teacher of Mathematics talked to us about Magic Squares.
The magic immediately caught us in it's mysterious arms, so we wanted to know more about this subject.
It is merely coincidence or is there another unknown power behind it?
Is it the same fascinating magic as we can also find in fractals?
Does nature have a secret that it will never reveal totally for mankind?
Is the entire universe with it's living creatures build up by a complicated (or simple ?!) pattern?
We wanted an answer on all these questions but actually … there are no answers at this moment.
So we decided to start with basic research of these intriguing constructions : Magic Squares.
What are they, do they have a use, how were they discovered, what will they learn us in the future ?
Let us take you on a journey through this magical world.
What Is A Magic Square?
A magic square is a simple mathematical game developed during the 1500s. You prepare a single square which is divided into the same number of rows and columns. Then, start filling each little square with the number from 1 to x where x equals the number of rows multiplied by the number of columns. You can only use a number once. Fill each square so that the sum of each row is the same as the sum of each column. In the example shown here, the sum of each row is 34, and the sum of each column is also 34. Note that the numbers from 1 through 16 is used only once. This is called a magic square.
Some of The Greatest Magic Squares
Take a look at this magic square. The sum of each row is 260, and so is the sum of each column. The numbers from 1 through 64 is used only once, so this is a valid magic square. But that's not all. When you add the first four numbers of any rows or columns, you get 130 which is the half of the sum of each row and column. But that's not the point! The greatest thing is that you can trace all the numbers, from 1 through 64, by "knight jumps". (A knight jump is a movement that a knight can perform in a chess game.) Try it. This amazing magic square was created at the beginning of the 18th century.
How about this magic square? Just like the previous example, the sum of each row is 260, and so is the sum of each column. The numbers from 1 through 64 is used only once, so this is a valid magic square. Like the previous example, when you add the first four numbers of any rows or colums, you get 130 which is the the half of the sum of each row and column. But from here is the different point than the previous one. Try adding any four little squares whose distances from the origin are the same. Also, take any squares consist of 4 little squares, and add those 4 numbers. In both cases, you get 130.
"The construction of magic squares is an amusement of great antiquity; we hear of magic squares in India and China before the Christian era, while they appear to have been introduced to Europe by Moscopulus who flourished Constantinople early in the fifteenth century.
"However, what was at first merely a practice of magicians and talisman makers has now for a long time become a serious study for mathematicians. Not that they have imagined that they would lead them to anything of solid advantage, but because the theory was seen to be fraught with difficulty, and it was considered possible that some new properties of numbers might be discovered which mathematicians could turn to account. This has in fact proved to be the case, for from a certain point of view the subject has been found to be algebraical rather than arithmetical and to be intimately connected with great departments of science such as the 'infinitesimal calculus,' the 'calculus of operations', and the theory of groups.
"No person living knows in how many ways it is possible to form a magic square of any order exceeding 4x4. The fact is that before we can attempt to enumerate magic squares we must see our way to solve problems of far more simple character.
"To say and to establish that problems of the general nature of the magic square are intimately connected with the infinitesimal calculus and the calculus of finite differences is to sum the matter up."
Magic cubes are extensions of some of the principles of magic squares. Magic squares are square arrays of numbers, in which the sum of each row column, and diagonal is identical. The sum of each row column and diagonal is called the magic number. The simplest magic square is a 3x3. There are countless ways to construct magic squares of even and odd order.
The characteristics of numeric magic cubes, odd or even, are that all straight columns, whether running from the top of the cube to the bottom, from the front to the back, or from one side to the other, should add up to the same number, and the four triagonals which unite of the cube also add up to the same number.
Magic squares and cubes are more than just number games; they use the principles of mathematics in their construction. Magic squares/cubes are included in the areas of: set theory, matrices, number theory, and combinatorics.
One might assume that some general formula exists such that all magic cubes and squares can be automatically produced by the computer, but there are countless and varying methods of construction. This field of study has continued to expand as computers are used to study the implications of number theory. Harold Stark of MIT included a chapter on "The Uniform Step Theory" (A method of constructing magic squares) in An Introduction to the Theory of Numbers, published in 1994. Because number theory is concerned with the properties of numbers, particularly integers, he considered the topic of theoretical interest.
The Rubiks Cube is sometimes called a magic cube. It has some similar properties; but it is constructed with colors rather than numbers. These magic cubes are not the topic of this project.
Magic cubes have several interesting properties. They are classified as either odd or even. In odd magic cubes, the sum of any two numbers diametrically opposite to each other should be equal to the sum of the first and the last series used. In these points the cube is close to perfection, but the corner diagonals of the 6 outside squares consist of various sums besides 42. Only one magic cube has been discovered with perfect characteristics, an 8x8x8. All others have perfect diagonals, sometimes called pandiagonals, or triagonals, in which the sums equal the magic number. On these cubes the diagonals on the faces of the cubes do not have the sum of the magic number. The smallest magic cube is 3x3x3. There are 27 numbers in this square. The magic number is 42.
Where do they come from?
Yin King is the name of a very old Chinese book. No one knows who has written it! In this book there is a story about a great turtle that appeared out of the Yellow river one day. On this great turtle's back were strange marks. These marks were dots that stood for the numbers 1 to 9. They were arranged in such way that no matter which direction you added the number up the answer was always 15. It was a magic square.
Japan was isolated from western society in the
Edo erra (16031867). During those days Japanese mathematician
created their own mathematical world. Many difficult problems
were presented and solved. Most of the answers were dedicated to
temples or shrines as beautiful pannels which were called
"SanGaku". Many of them were lost during the tide of
modernization after the Meiji revolution. About nine hundred
SanGaku, howver, are seen nowadays in rural area of Japan.
Many books on the Japanese mathematics (called
"WaSan") were also published during the
TokugawaShogun's period.
These mathematics are called "WaSan"(old Japanese
Math) in order to distinguish them from western mathematics.
Beauty of the WaSan was recently introduced to the western
society by "Japanese Temple Geometry Problems" written
by Mr. H. Fukagawa, published from Charles Babbage Research
Centre (1989).
Mr. Fukagawa wrote another book on WaSan (not translated into
English yet). Following short comments on the magic squares in
Japan were seen in the book.
2 1 3 3 2 1 1 3 2
Above magic square using 1 , 2 and 3 was seen in a book publised in 1743.
6 1 8 7 5 3 2 9 4
This well known square was seen in another book published in 1840.
19 16 17 13 18 14 15 11 12 1 9 10 6 7 3 8 4 5 2
This magic circle was reported in 1660.
World Records from 1975 to now
1975 105 x 105 Richard Suntag (Pomona, USA)
1979 501 x 501 Gerolf Lenz ( Wuppertal, Germany)
1987 897 x 897 Frank Tast and Uli Schmidt ( Pforzheim, Germany)
1988 1000 x 1000 Christian Schaller ( Munich, Germany)
1989 2001 x 2001 Sven Paulus and Ralph Bülling and Jörg Sutter
( Pforzheim, Germany)
1991 2121 x 2121 Ralf Laue (Leipzig, Germany)
1994 3001 x 3001 Louis Caya ( SainteFoy, Canada)
Largest magic square written by hand
1990 1111 x 1111 Norbert Behnke ( Krefeld, Germany)
The following 25x25 matrix is a magic square.It has the following properties:
If one squares all entries in in the square, the square remains magic: all row,column and diagonal sums are equal to 3263025
A very, very magic square
1 
443 
235 
547 
339 
283 
100 
387 
179 
616 
565 
352 
44 
456 
148 
217 
509 
321 
113 
405 
499 
161 
578 
270 
57 

157 
599 
261 
53 
495 
439 
226 
543 
335 
22 
91 
383 
200 
612 
279 
373 
40 
452 
144 
556 
505 
317 
109 
421 
213 

313 
105 
417 
209 
521 
595 
257 
74 
486 
153 
247 
539 
326 
18 
435 
379 
191 
608 
300 
87 
31 
473 
140 
552 
369 

469 
131 
573 
365 
27 
121 
413 
205 
517 
309 
253 
70 
482 
174 
586 
535 
347 
14 
426 
243 
187 
604 
291 
83 
400 

625 
287 
79 
391 
183 
127 
569 
356 
48 
465 
409 
221 
513 
305 
117 
61 
478 
170 
582 
274 
343 
10 
447 
239 
526 

587 
254 
66 
483 
175 
244 
531 
348 
15 
427 
396 
188 
605 
292 
84 
28 
470 
132 
574 
361 
310 
122 
414 
201 
518 

118 
410 
222 
514 
301 
275 
62 
479 
166 
583 
527 
344 
6 
448 
240 
184 
621 
288 
80 
392 
461 
128 
570 
357 
49 

149 
561 
353 
45 
457 
401 
218 
510 
322 
114 
58 
500 
162 
579 
266 
340 
2 
444 
231 
548 
617 
284 
96 
388 
180 

280 
92 
384 
196 
613 
557 
374 
36 
453 
145 
214 
501 
318 
110 
422 
491 
158 
600 
262 
54 
23 
440 
227 
544 
331 

431 
248 
540 
327 
19 
88 
380 
192 
609 
296 
370 
32 
474 
136 
553 
522 
314 
101 
418 
210 
154 
591 
258 
75 
487 

423 
215 
502 
319 
106 
55 
492 
159 
596 
263 
332 
24 
436 
228 
545 
614 
276 
93 
385 
197 
141 
558 
375 
37 
454 

554 
366 
33 
475 
137 
206 
523 
315 
102 
419 
488 
155 
592 
259 
71 
20 
432 
249 
536 
328 
297 
89 
376 
193 
610 

85 
397 
189 
601 
293 
362 
29 
466 
133 
575 
519 
306 
123 
415 
202 
171 
588 
255 
67 
484 
428 
245 
532 
349 
11 

236 
528 
345 
7 
449 
393 
185 
622 
289 
76 
50 
462 
129 
566 
358 
302 
119 
406 
223 
515 
584 
271 
63 
480 
167 

267 
59 
496 
163 
580 
549 
336 
3 
445 
232 
176 
618 
285 
97 
389 
458 
150 
562 
354 
41 
115 
402 
219 
506 
323 

359 
46 
463 
130 
567 
511 
303 
120 
407 
224 
168 
585 
272 
64 
476 
450 
237 
529 
341 
8 
77 
394 
181 
623 
290 

390 
177 
619 
281 
98 
42 
459 
146 
563 
355 
324 
111 
403 
220 
507 
576 
268 
60 
497 
164 
233 
550 
337 
4 
441 

541 
333 
25 
437 
229 
198 
615 
277 
94 
381 
455 
142 
559 
371 
38 
107 
424 
211 
503 
320 
264 
51 
493 
160 
597 

72 
489 
151 
593 
260 
329 
16 
433 
250 
537 
606 
298 
90 
377 
194 
138 
555 
367 
34 
471 
420 
207 
524 
311 
103 

203 
520 
307 
124 
411 
485 
172 
589 
251 
68 
12 
429 
241 
533 
350 
294 
81 
398 
190 
602 
571 
363 
30 
467 
134 

195 
607 
299 
86 
378 
472 
139 
551 
368 
35 
104 
416 
208 
525 
312 
256 
73 
490 
152 
594 
538 
330 
17 
434 
246 

346 
13 
430 
242 
534 
603 
295 
82 
399 
186 
135 
572 
364 
26 
468 
412 
204 
516 
308 
125 
69 
481 
173 
590 
252 

477 
169 
581 
273 
65 
9 
446 
238 
530 
342 
286 
78 
395 
182 
624 
568 
360 
47 
464 
126 
225 
512 
304 
116 
408 

508 
325 
112 
404 
216 
165 
577 
269 
56 
498 
442 
234 
546 
338 
5 
99 
386 
178 
620 
282 
351 
43 
460 
147 
564 

39 
451 
143 
560 
372 
316 
108 
425 
212 
504 
598 
265 
52 
494 
156 
230 
542 
334 
21 
438 
382 
199 
611 
278 
95 
These squares are complete and selfsimilar
square and have special constellation patterns all at a time !
(1) complete property;
In the squares,not only the major but olso the minor diagonals
add up to the same number 65.
For examples;
A * * * * * B * * * * * C * * * * * D * A + B + C + D + E = 65 (Major diagonal) * * * * E * A * * * * * B * * * * * C * * * * * D A + B + C + D + E = 65 E * * * * * * A * * * * * B * * * * * C D * * * * A + B + C + D + E = 65 * E * * * ... ... and so on.
(2) Self complement;
The square is invariant for the complemental transform. If you
change all the number "n" of the square by
"26n" , you get the rotated same square.
You can say that it is axisymmetry or selfsimillar, in another
word.
(3) fivestar constellation patterns;
Sum of the five numbers of all small squares and the centers sum
to the same number 65.
1 * * * 9 1 * 22 * * * 15 * 18 * * * 22 * 9 * * * * * * 19 * * * * * 6 * * * * * 5 * * * 13 * * 10 * 13 * * * 2 * 24 * * * 13 * 16 * * * * * * * * * * * * * * * * * * * * 17 * * * 25 * * * * * * * * * * * * * * * .... .... * * * * * * * * * * * * 13 * 16 .... .... .... .... .... * * * 7 * * * 4 * 25 Rhonbohedral patterns; Furthermore, five numbers of all small rhonbohedrons add up to the same number 65. * 15 * * * * * 22 * * 23 19 6 * * * 19 6 5 * * 2 * * * * * 13 * * * * * * * * * * * * * * * * * * * * * * ...... ...... ...... * * * * * * * 22 * * * * * * * * * * * * * * * 24 * 10 * 13 * 16 * * 20 7 3 * * * * * * * * 11 * * * 4 * *
( 1) 1 15 22 18 9 23 19 6 5 12 10 2 13 24 16 14 21 20 7 3 17 8 4 11 25 
( 2) 1 15 24 18 7 23 17 6 5 14 10 4 13 22 16 12 21 20 9 3 19 8 2 11 25 
( 3) 1 23 20 12 9 15 7 4 21 18 24 16 13 10 2 8 5 22 19 11 17 14 6 3 25 
( 4) 1 23 20 14 7 15 9 2 21 18 22 16 13 10 4 8 5 24 17 11 19 12 6 3 25 
( 5) 2 14 21 18 10 23 20 7 4 11 9 1 13 25 17 15 22 19 6 3 16 8 5 12 24 
( 6) 2 14 25 18 6 23 16 7 4 15 9 5 13 21 17 11 22 19 10 3 20 8 1 12 24 
( 7) 2 23 19 11 10 14 6 5 22 18 25 17 13 9 1 8 4 21 20 12 16 15 7 3 24 
( 8) 2 23 19 15 6 14 10 1 22 18 21 17 13 9 5 8 4 25 16 12 20 11 7 3 24 
( 9) 4 12 21 18 10 23 20 9 2 11 7 1 13 25 19 15 24 17 6 3 16 8 5 14 22 
(10) 4 12 25 18 6 23 16 9 2 15 7 5 13 21 19 11 24 17 10 3 20 8 1 14 22 
(11) 4 23 17 11 10 12 6 5 24 18 25 19 13 7 1 8 2 21 20 14 16 15 9 3 22 
(12) 4 23 17 15 6 12 10 1 24 18 21 19 13 7 5 8 2 25 16 14 20 11 9 3 22 
(13) 5 11 22 18 9 23 19 10 1 12 6 2 13 24 20 14 25 16 7 3 17 8 4 15 21 
(14) 5 11 24 18 7 23 17 10 1 14 6 4 13 22 20 12 25 16 9 3 19 8 2 15 21 
(15) 5 23 16 12 9 11 7 4 25 18 24 20 13 6 2 8 1 22 19 15 17 14 10 3 21 
(16) 5 23 16 14 7 11 9 2 25 18 22 20 13 6 4 8 1 24 17 15 19 12 10 3 21 
Note :
We found most of the information on Magic Squares on the Internet.
Next are the internet URL's (Uniform Resource Locator) where you can find more info.
http://www.indiaheritage.com/magicsq.htm
http://www.pse.che.tohoku.ac.jp/~msuzuki/MagicSquare.5x5.selfsim.html
http://www.amersol.edu.pe/highschool/magic.cgi
http://www.inetworld.net/~houlton/
http://www.imn.htwkleipzig.de/~saxonia/records/magic.html
http://www.math.unibas.ch/~hderksen/magic.html
http://forum.swarthmore.edu/alejandre/magic.square/ben.html
http://users.powernet.co.uk/higgs/f3_sun.html