886:
478:
In the antimagic square of order 5 on the left, the rows, columns and diagonals sum up to numbers between 60 and 71. In the antimagic square on the right, the rows, columns and diagonals add up to numbers in the range 59–70.
589:. (Thus, they are the relaxation in which no particular values are required for the row, column, and diagonal sums.) There are no heterosquares of order 2, but heterosquares exist for any order
564:, and whose row-sums and column-sums constitute a set of consecutive integers. If the diagonals are included in the set of consecutive integers, the array is known as a
562:
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282:
In both of these antimagic squares of order 4, the rows, columns and diagonals sum to ten different numbers in the range 29–38.
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in a square, such that the rows, columns, and diagonals all sum to different values has been called a
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in order, then exchanging 1 and 2. It is suspected that there are exactly 3120
47:. The smallest antimagic squares have order 4. Antimagic squares contrast with
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of nonnegative integers whose nonzero entries are the consecutive integers
754:
728:"Sparse anti-magic squares and vertex-magic labelings of bipartite graphs"
44:
602:
51:, where each row, column, and diagonal sum must have the same value.
568:(STAM). Note that a STAM is not necessarily a SAM, and vice versa.
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613:, a heterosquare results from writing the numbers 1 to
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39:columns and the two diagonals form a sequence of 2
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682:: CS1 maint: multiple names: authors list (
605:pattern will produce a heterosquare, and if
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31:in a square, such that the sums of the
726:Gray, I. D.; MacDougall, J.A. (2006).
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27:is an arrangement of the numbers 1 to
7:
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653:
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491:(SAM) is a square matrix of size
884:
566:sparse totally anti-magic square
1:
926:Prime reciprocal magic square
797:Peter Bartsch's Heterosquares
581:square with the numbers 1 to
557:{\displaystyle m\leq n^{2}}
524:{\displaystyle 1,\ldots ,m}
43: + 2 consecutive
1102:
745:10.1016/j.disc.2006.04.032
621:heterosquares of order 3.
601:, filling the square in a
940:Higher dimensional shapes
931:Most-perfect magic square
882:
286:Order 5 antimagic squares
176:
70:
60:Order 4 antimagic squares
985:Pandiagonal magic square
980:Associative magic square
921:Pandiagonal magic square
489:sparse antimagic square
558:
525:
707:www.magic-squares.net
664:mathworld.wolfram.com
658:W., Weisstein, Eric.
619:essentially different
559:
526:
799:at magic-squares.net
732:Discrete Mathematics
703:"Anti-magic Squares"
535:
503:
1021:Eight queens puzzle
593: ≥ 3: if
16:Mathematical object
1062:"Antimagic Square"
1059:Weisstein, Eric W.
774:Weisstein, Eric W.
660:"Antimagic Square"
554:
521:
1044:
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990:Multimagic square
901:Alphamagic square
738:(22): 2878–2892.
571:A filling of the
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999:Related concepts
906:Antimagic square
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21:antimagic square
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1016:Number Scrabble
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964:Magic hyperbeam
959:Magic hypercube
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911:Geomagic square
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483:Generalizations
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1051:External links
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1026:Magic constant
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973:Classification
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894:Related shapes
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876:Magic triangle
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861:Magic hexagram
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837:Magic polygons
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777:"Heterosquare"
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755:1959.13/803634
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1086:Magic squares
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856:Magic hexagon
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49:magic squares
46:
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34:
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1036:Magic series
1006:Latin square
916:Heterosquare
905:
866:Magic square
851:Magic circle
792:
780:
735:
731:
721:
710:. Retrieved
706:
667:. Retrieved
663:
636:J. A. Lindon
631:Magic square
614:
606:
594:
590:
587:heterosquare
586:
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281:
40:
36:
32:
28:
24:
20:
18:
1031:Magic graph
1011:Word square
245:→ 36
228:→ 33
211:→ 38
194:→ 29
139:→ 31
122:→ 32
105:→ 38
88:→ 35
947:Magic cube
871:Magic star
712:2016-12-03
669:2016-12-03
642:References
272: 31
35:rows, the
1067:MathWorld
782:MathWorld
542:≤
531:for some
513:…
179:32
166: 29
73:34
23:of order
1080:Category
678:cite web
625:See also
55:Examples
45:integers
952:classes
603:spiral
844:Types
684:link
611:even
750:hdl
740:doi
736:306
609:is
599:odd
597:is
495:by
468:10
465:22
459:12
456:14
451:25
445:19
439:15
431:11
428:23
425:20
417:24
414:16
411:13
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388:21
377:16
371:17
368:14
365:12
357:24
351:18
348:11
343:25
340:15
331:21
323:10
320:13
317:23
314:19
309:22
303:20
267:35
262:34
257:30
252:37
161:36
156:37
151:30
146:33
19:An
1082::
1064:.
779:.
764:^
748:.
734:.
730:.
705:.
692:^
680:}}
676:{{
662:.
650:^
576:×
487:A
462:9
448:2
442:8
434:1
422:5
408:3
405:7
400:4
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374:4
360:1
354:7
337:3
334:6
326:2
306:9
300:8
297:5
239:11
233:14
222:16
208:10
199:15
191:12
185:13
136:10
133:11
116:14
102:12
93:16
85:13
79:15
1070:.
829:e
822:t
815:v
785:.
758:.
752::
742::
715:.
686:)
672:.
615:n
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595:n
591:n
583:n
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574:n
550:2
546:n
539:m
519:m
516:,
510:,
507:1
497:n
493:n
270:↘
265:↓
260:↓
255:↓
250:↓
242:5
236:6
225:8
219:2
216:7
205:4
202:9
188:3
182:1
177:↙
164:↘
159:↓
154:↓
149:↓
144:↓
130:4
127:6
119:1
113:8
110:9
99:7
96:3
82:5
76:2
71:↙
41:n
37:n
33:n
29:n
25:n
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