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20:
93:
138:
squares using complex numbers, a ploy leading to a new theorem that correlated every 3 × 3 magic square with a unique parallelogram on the complex plane. Continuing in the same vein, a decisive next step was to interpret the variables in the Lucas formula as standing for geometrical forms, an outlandish idea that led directly to the concept of a geomagic square. It turned out to be an unexpected consequence of this find that traditional magic squares now became revealed as one-dimensional geomagic squares.
205:
708:
1136:
696:
105:
538:
197:—which is to say, are also target-tiling triads. If so, a 3 × 3 geomagic square using 9 decominoes and selected target has been identified. If this fails, alternative target shapes can be tried. An elaborated version of the same method can be used to search for larger squares, or for squares including differently-sized pieces.
565:. In this case the target "shape" for the geomagic square at right is simply a one dimensional line segment 15 units long, the pieces again being no more than straight line segments. As such, the latter is obviously a straightforward translation into geometrical terms of the numerical magic square at left.
121:, or composed of separated islands, as seen in Figure 3. Since they can be placed so as to mutually overlap, disjoint pieces are often able to tile areas that connected pieces cannot. The rewards of this extra pliancy are often to be seen in geomagics that possess symmetries denied to numerical specimens.
242:
548:
Contrary to the impression made at first sight, it is a misunderstanding to regard the term 'geomagic square' as referring to some category of magic square. In fact the exact opposite is the case: every (additive) magic square is a particular instance of a geomagic square, but never vice versa. The
192:
will consist of a list of integer triads. A subsequent routine can then run through and test every combination of three different triads in turn. The test will consist in treating the candidate triads as the row entries in a 3 × 3 square, and then checking to see whether the columns and
691:, shares the same magic property as the rows and columns. However, it is easily shown that a panmagic square of size 3 × 3 is impossible to construct with numbers, whereas a geometric example can be seen in Figure 3. No comparable example using connected pieces has yet been reported.
87:
Surprisingly, computer investigations show that Figure 2 is just one among 4,370 distinct 3 × 3 geomagic squares using pieces with these same sizes and same target. Conversely, Figure 1 is one of only two solutions using similar-sized pieces and identical target. In general, repeated
200:
An alternative method of construction begins with a trivial geomagic square showing repeated pieces, the shapes of which are then modified so as to render each distinct, but without disrupting the square's magic property. This is achieved by means of an algebraic template such as seen below, the
165:
said, "To come up with this after thousands of years of study of magic squares is pretty amazing." It may be asked whether geomagic squares might have applications outside the study of puzzles. Cameron is convinced of it, saying, "I can immediately see a lot of things I'd like to do with this."
137:
characterizes the structure of every 3 × 3 magic square of numbers. Sallows, already the author of original work in this area, had long speculated that the Lucas formula might contain hidden potential. This surmise was confirmed in 1997 when he published a short paper that examined
666:
The point being that every numerical magic square can be understood as a one-dimensional geomagic square as above. Or as
Sallows himself puts it, "Traditional magic squares featuring numbers are then revealed as that particular case of 'geomagic' squares in which the elements are all
738:
A second example is Figure 4, which is a so-called 'self-interlocking' geomagic square. Here the 16 pieces are no longer contained within separate cells, but define the square cell shapes themselves, so as to mesh together to complete a square-shaped jigsaw.
533:{\displaystyle {\begin{array}{|c|c|c|c|}\hline k+a+b&k-a+d&k-c-d&k-b+c\\\hline k+a-b&k-a-d&k-c+d&k+b+c\\\hline k-a-b&k+a-d&k+c+d&k+b-c\\\hline k-a+b&k+a+d&k+c-d&k-b-c\\\hline \end{array}}}
63:. As with numerical types, it is required that the entries in a geomagic square be distinct. Similarly, the eight trivial variants of any square resulting from its rotation and/or reflection are all counted as the same square. By the
662:
As
Delahaye says, "This example shows that the geomagic square concept generalizes magic squares. The result here is hardly spectacular, but happily there are other geomagic squares that are not the result of such a translation."
964:
Cet exemple montre que la notion de carré géomagique généralise celle de carré magique. Le résultat n’est ici guère spectaculaire, mais heureusement, il existe d’autres carrés géomagiques ne provenant pas d’une telle traduction
183:
In the case of Figure 1, for instance, a first step would be to decide on the piece sizes to be used (in this case all the same), and the shape of the desired target. An initial program would then be able to generate a list
124:
Besides squares using planar shapes, there exist 3D specimens, the cells of which contain solid pieces that will combine to form the same constant solid target. Figure 5 shows an example in which the target is a cube.
75:
Figure 1 above shows a 3 × 3 geomagic square. The 3 pieces occupying each row, column and diagonal pave a rectangular target, as seen at left and right, and above and below. Here the 9 pieces are all
720:
In addition to being geomagic, there exist squares with auxiliary properties making them even more distinctive. In Figure 6, for example, which is magic on rows and columns only, the 16 pieces form a so-called
59:. A geomagic square, on the other hand, is a square array of geometrical shapes in which those appearing in each row, column, or diagonal can be fitted together to create an identical shape called the
671:
line segments, and which do not correspond to any numerical magic square. Thus, even in dimension one, the traditional types correspond to only a tiny subset of all geometric magic squares.
67:
of a geomagic square is meant the dimension of the pieces it uses. Hitherto interest has focused mainly on 2D squares using planar pieces, but pieces of any dimension are permitted.
1005:
161:, called geomagic squares "a wonderful new piece of recreational maths, which will delight non-mathematicians and give mathematicians food for thought." Mathematics writer
247:
679:
The richer structure of geomagic squares is reflected in the existence of specimens showing a far greater degree of 'magic' than is possible with numerical types. Thus a
188:
corresponding to every possible tiling of this target shape by 3 distinct decominoes (polyominoes of size 10). Each decomino is represented by a unique integer, so that
174:
Trivial examples excepted, there are no known easy methods for producing geomagic squares. To date, two approaches have been explored. Where the pieces to be used are
201:
distinct variables in which are then interpreted as different shapes to be either appended to or excised from the initial pieces, depending on their sign.
142:
236:
represent the protrusions (+) and/or indentations (-) by means of which it becomes modified so as to result in 16 distinct jigsaw pieces.
80:, but pieces of any shape may appear, and it is not a requirement that they be of same size. In Figure 2, for instance, the pieces are
918:
88:
piece sizes imply fewer solutions. However, at present there exists no theoretical underpinning to explain these empirical findings.
1077:
990:
667:
one-dimensional." This however does not exhaust the 1D case, because there exist 1D geomagic squares whose components are
948:
903:
146:
799:
763:. The stamp below, showing one of the geomagic squares created by Sallows, was chosen to be in this collection.
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of consecutive sizes from 1 up to 9 units. The target is a 4 × 4 square with an inner square hole.
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point is made clear by the example below that appears in a wide-ranging article on geomagic squares by
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180:, or shapes built up from repeated units, an exhaustive search by computer becomes possible.
92:
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8:
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51:
in 2001. A traditional magic square is a square array of numbers (almost always positive
1026:
Geometric Magic
Squares: A Challenging New Twist Using Colored Shapes Instead of Numbers
1201:
731:
distinct shapes, each of which can be tiled by smaller replicas of the complete set of
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Figure 4 illustrates such a geometrical interpretation of the template in which
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55:) whose sum taken in any row, any column, or in either diagonal is the same
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Complex
Projective 4-Space Where exciting things happen: Geomagic squares
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149:, speaks of the field of magic squares being "dramatically expanded"
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28:
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Macau's magic square stamps just made philately even more nerdy
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diagonals thus formed each contain 3 integers that are also in
715:
A geomagic square whose pieces comprise a self-tiling tile set
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839:"Alphamagic Squares", thinkquest.org:Magic of Mathematics
787:
by Jean-Paul
Delahaye, by Jean-Paul Delahaye, 04-07-2013
685:
is one in which every diagonal, including the so-called
543:
850:"New advances with 4 × 4 magic squares" by Lee Sallows
742:
245:
1051:
A formal mathematical definition of
Geomagic Squares
919:"Ancient puzzle gets new lease of 'geomagical' life"
100:
A geomagic square using consecutively-sized pieces.
532:
1298:
944:
942:
914:
912:
800:"Magic squares are given a whole new dimension"
220:is interpreted as a small square shape, while
153:, winner of the London Mathematical Society's
133:A well-known formula due to the mathematician
1071:
939:
703:A 3D geomagic square with cubic target shapes
909:
751:Macau stamp featuring geometric magic square
117:The pieces in a geomagic square may also be
112:A panmagic 3 × 3 geomagic square
1078:
1064:
795:
793:
169:
27:A geomagic square with same-sized pieces (
974:On Self-Tiling Tile Sets by Lee Sallows,
818:
816:
814:
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694:
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103:
91:
18:
790:
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823:Geometric Magic Squares by Lee Sallows
811:
755:On October 9, 2014 the post office of
727:. Such a set is defined as any set of
1059:
829:, Vol 23, No. 4 Winter 2011, pp 25-31
544:Relation to traditional magic squares
212:A 'self-interlocking' geomagic square
141:Other researchers also took notice.
904:Mathematical Association of America
759:issued a series of stamps based on
743:Geomagic squares in popular culture
147:Journal of Recreational Mathematics
13:
1085:
1028:, Dover Publications, April 2013,
16:Form of magic squares using shapes
14:
1328:
1039:
870:"The Lost Theorem" by Lee Sallows
1134:
949:Les carrés magiques géométriques
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618:Target is •••••••••••••••
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874:The Mathematical Intelligencer
863:
854:
843:
832:
827:The Mathematical Intelligencer
778:
1:
1176:Prime reciprocal magic square
901:reviewed by Charles Ashbacher
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877:Vol 19, No. 4, pp 51-4, 1997
7:
70:
10:
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988:Macau Post Office web site
128:
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1190:Higher dimensional shapes
1189:
1181:Most-perfect magic square
1143:
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1093:
1011:Science, November 3, 2014
43:, is a generalization of
1235:Pandiagonal magic square
1230:Associative magic square
1171:Pandiagonal magic square
1046:Geomagic Squares website
785:Hidden geometric nuggets
771:
559:, the French version of
157:and joint winner of the
951:by Jean-Paul Delahaye,
899:Geometric Magic Squares
170:Methods of construction
39:, often abbreviated to
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101:
37:geometric magic square
32:
750:
710:
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107:
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22:
1317:Geometric dissection
976:Mathematics Magazine
906:, September 24, 2013
860:Sallows, pp 3 and 91
724:self-tiling tile set
243:
1271:Eight queens puzzle
619:
575:
562:Scientific American
145:, co-editor of the
993:2014-11-11 at the
955:No. 428, June 2013
927:, January 24, 2011
802:, by Alex Bellos,
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551:Jean-Paul Delahaye
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1240:Multimagic square
1151:Alphamagic square
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143:Charles Ashbacher
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1249:Related concepts
1156:Antimagic square
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1209:Magic hypercube
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1161:Geomagic square
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953:Pour La Science
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921:by Jacob Aron,
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682:panmagic square
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556:Pour la Science
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41:geomagic square
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1223:Classification
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1126:Magic triangle
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1111:Magic hexagram
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1087:Magic polygons
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1040:External links
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1024:Sallows, Lee,
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1307:Magic squares
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675:Special types
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135:Édouard Lucas
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57:target number
54:
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45:magic squares
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1286:Magic series
1256:Latin square
1166:Heterosquare
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1116:Magic square
1101:Magic circle
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1009:The Guardian
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61:target shape
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47:invented by
40:
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34:
1281:Magic graph
1261:Word square
163:Alex Bellos
159:Euler Medal
82:polyominoes
49:Lee Sallows
1301:Categories
1197:Magic cube
1121:Magic star
1034:0486489094
767:References
711:Figure 6:
699:Figure 5:
208:Figure 4:
108:Figure 3:
96:Figure 2:
78:decominoes
29:decominoes
23:Figure 1:
626:•••••••••
521:−
515:−
504:−
464:−
451:−
417:−
400:−
394:−
358:−
347:−
341:−
330:−
305:−
294:−
288:−
271:−
177:polyforms
65:dimension
991:Archived
965:directe.
735:shapes.
645:••••••••
640:•••••••
119:disjoint
71:Examples
53:integers
1202:classes
1019:Sources
651:••••••
129:History
1032:
713:
701:
210:
110:
98:
25:
1094:Types
772:Notes
757:Macau
637:•••••
1030:ISBN
623:••••
232:and
634:•••
629:••
553:in
1303::
941:^
911:^
825:,
813:^
792:^
607:6
596:7
585:2
228:,
224:,
35:A
1079:e
1072:t
1065:v
733:n
729:n
648:•
604:1
601:8
593:5
590:3
582:9
579:4
524:c
518:b
512:k
507:d
501:c
498:+
495:k
490:d
487:+
484:a
481:+
478:k
473:b
470:+
467:a
461:k
454:c
448:b
445:+
442:k
437:d
434:+
431:c
428:+
425:k
420:d
414:a
411:+
408:k
403:b
397:a
391:k
384:c
381:+
378:b
375:+
372:k
367:d
364:+
361:c
355:k
350:d
344:a
338:k
333:b
327:a
324:+
321:k
314:c
311:+
308:b
302:k
297:d
291:c
285:k
280:d
277:+
274:a
268:k
263:b
260:+
257:a
254:+
251:k
234:d
230:c
226:b
222:a
218:k
195:L
190:L
186:L
31:)
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