923:
963:
943:
122:
169:
25:
470:
463:
456:
449:
442:
432:
425:
415:
380:
373:
366:
359:
349:
342:
332:
325:
290:
283:
276:
266:
259:
249:
242:
235:
1064:
Another puzzle with similar properties is MacMahon's Cubes, which are a set of 30 cubes, with sides colored one of 6 different colors. Unlike the MacMahon
Squares puzzle, we do not include all 2,226 possible cubes, but only the cubes containing exactly 6 distinct colors and 1 of each of the 6 colors.
1101:
The nomenclature in this table is intended to be orientation-agnostic, starting from the edge(s) with the lowest value and proceeding anti-clockwise. For example, the tile 1232 has the lowest color value 1, followed sequentially by 2 (immediately anti-clockwise of color value 1), then 3, and finally
1025:
Another way to change the puzzle is to restrict which colors squared make up the border colors. In the classic MacMahon squares puzzle, there are a total of 20 places on the border. The number of each color that can be present on these 20 places can be described by
994:. Here, a, b, and c represent the shift in colors to which the first, second, and third colors can be matched to. A '1' it is matched to itself, and a '2' signifies that it must be matched with a different color.
1005:
represents 1 to 1 and 2 to 3 as the 1 to 1 matching is represented by the number 1, and the matching between 2 and 3 is represented by 2. More colors can be described in a similar way. For example, a coloring of
849:
1048:
From here we can see that the only possible numbers to describe the number of each color composing the boundary are even numbers, since this would imply an odd number of another color, which would violate the
618:
1013:
From here we can see that the only possible numbers to describe the pairings by are 1 and 2, since a 3 or above merely skips over a color that would be used the same otherwise because colorings are
160:
puzzles have multiple variants, which are determined by restrictions on how to arrange the 24 squares. This game has also been commercialized in numerous physical forms, by various companies.
903:
751:
697:
188:. The original version consisted of one copy of each of the 24 different squares that can be made by coloring the edges of a square with one of three colors. (Here "different" means
641:
542:
192:
rotations.) The goal is to arrange the squares into a 4 by 6 grid so that when two squares share an edge, the common edge is the same color in both squares.
148:
with 3-color patterns; each of the four edges is assigned a single color. The complete set of 24 squares are organized next to each other by matching edge
990:
suggested the ability to define which borders can contact one another, based on their colors. This is by some permutation of the 3 colors, described by C
521:), then progressing in detail with similar games using tiles shaped as equilateral triangles, squares, right isoceles triangles, cubes, and hexagons.
922:
129:
constraint which forces a single color along the perimeter, while also maximizing the largest contiguous single-color area within the perimeter
1037:
represents 20 of the first color and none of the rest since the first color already constitutes all available border spaces. Alternatively, B
962:
942:
758:
908:
For example, given a triangle with three sides, each of which is assigned one of four possible colors, there are 56 unique patterns.
108:
42:
1041:
represents 10 of the first color and 10 of the next. More colors can be described in a similar way. For example, a boundary of B
547:
89:
61:
46:
199:
was used to produce 12,261 solutions to the basic version of the MacMahon
Squares puzzle, with a runtime of about 40 hours.
68:
1045:
represents 22 of the first, 16 of the second, 8 of the next, and 2 of the last colors to populate the border colors.
1068:
MacMahon
Squares have served as a baseline for numerous other puzzles. Some of these include the Nelson Puzzle, the
1001:
represents 1 to 1, 2 to 2, and 3 to 3, as each of these matchings are represented by the number 1. Alternatively, C
856:
704:
650:
75:
35:
1177:
1149:
987:
181:
141:
57:
1014:
1050:
137:
1220:
509:
The MacMahon
Squares game is an example of an edge-matching puzzle. The family of such problems is
82:
1201:
1262:
1085:, patented in 1969 by Marc Odier, uses triangular tiles first proposed by MacMahon in 1921.
121:
510:
524:
There are a total of 24 distinct squares for 3 colors. For an arbitrary number of colors
1061:
MacMahon
Squares, along with variations on the idea, was commercialized as Multimatch.
626:
527:
1256:
196:
157:
1153:
24:
168:
153:
1069:
1111:
With three colors and four edges, one color will be repeated at least once.
469:
462:
455:
448:
441:
431:
424:
414:
379:
372:
365:
358:
348:
341:
331:
324:
289:
282:
275:
265:
258:
248:
241:
234:
1073:
1081:
1030:
where a, b, and c are the number of each color of the border pieces.
518:
145:
844:{\displaystyle {\frac {n}{6}}\cdot (n+1)\cdot (n^{4}-n^{3}+n^{2}+2)}
189:
167:
149:
120:
1200:
Gardner, Martin. "The 24 Color
Squares and the 30 Color Cubes".
643:
colors, MacMahon determined the number of unique patterns are:
544:, the number of unique squares can be found by the expression
517:
describes these games in general, starting with linear forms (
18:
1247:
613:{\displaystyle {\frac {n}{4}}\cdot (n+1)\cdot (n^{2}-n+2)}
172:
All 24 MacMahon
Squares, redrawn from Fig. 26 of
859:
761:
707:
653:
629:
550:
530:
49:. Unsourced material may be challenged and removed.
897:
843:
745:
691:
635:
612:
536:
1010:represents 1 to 1, 2 to 3, 4 to 5, and 6 to 7.
898:{\displaystyle {\frac {n}{7}}\cdot (n^{6}+6)}
746:{\displaystyle {\frac {n}{5}}\cdot (n^{4}+4)}
692:{\displaystyle {\frac {n}{3}}\cdot (n^{2}+2)}
125:A particular solution; this illustrates the B
8:
468:
461:
454:
447:
440:
430:
423:
413:
378:
371:
364:
357:
347:
340:
330:
323:
288:
281:
274:
264:
257:
247:
240:
233:
16:Puzzle published in 1921 by Percy MacMahon
880:
860:
858:
826:
813:
800:
762:
760:
728:
708:
706:
674:
654:
652:
628:
589:
551:
549:
529:
109:Learn how and when to remove this message
201:
180:MacMahon squares was first published in
1124:
1094:
915:
7:
1195:
1193:
1144:
1142:
1140:
1138:
1136:
1134:
1132:
1130:
1128:
47:adding citations to reliable sources
1221:"Age of Puzzles - Harry L. Nelson"
1053:of the total number of triangles.
14:
917:MacMahon Squares sample solutions
961:
941:
921:
23:
34:needs additional citations for
892:
873:
838:
793:
787:
775:
740:
721:
686:
667:
607:
582:
576:
564:
1:
1158:. Cambridge University Press
203:MacMahon squares (3 colors)
1203:New Mathematical Diversions
515:New Mathematical Diversions
1279:
1178:"MacMahon 3-Color Squares"
1079:The commercial board game
1155:New Mathematical Pastimes
436:
419:
353:
336:
270:
253:
227:
224:
221:
213:
210:
186:New Mathematical Pastimes
174:New Mathematical Pastimes
144:in 1921, using 24 unique
182:Percy Alexander MacMahon
1248:MacMahon Squares Online
899:
845:
747:
693:
637:
623:For other shapes with
614:
538:
177:
130:
900:
846:
748:
694:
638:
615:
539:
171:
124:
1225:www.ageofpuzzles.com
857:
759:
705:
651:
627:
548:
528:
513:. The first part of
138:edge-matching puzzle
43:improve this article
204:
152:to create a 4 by 6
140:first published by
895:
841:
743:
689:
633:
610:
534:
202:
178:
131:
58:"MacMahon Squares"
1057:Analogous puzzles
868:
770:
716:
662:
636:{\displaystyle n}
559:
537:{\displaystyle n}
502:
501:
184:'s 1921 treatise
119:
118:
111:
93:
1270:
1235:
1234:
1232:
1231:
1217:
1211:
1210:
1208:
1197:
1188:
1187:
1185:
1184:
1174:
1168:
1167:
1165:
1163:
1146:
1112:
1109:
1103:
1099:
965:
945:
925:
904:
902:
901:
896:
885:
884:
869:
861:
850:
848:
847:
842:
831:
830:
818:
817:
805:
804:
771:
763:
752:
750:
749:
744:
733:
732:
717:
709:
698:
696:
695:
690:
679:
678:
663:
655:
642:
640:
639:
634:
619:
617:
616:
611:
594:
593:
560:
552:
543:
541:
540:
535:
472:
465:
458:
451:
444:
434:
427:
417:
382:
375:
368:
361:
351:
344:
334:
327:
292:
285:
278:
268:
261:
251:
244:
237:
205:
176:(MacMahon, 1921)
134:MacMahon Squares
114:
107:
103:
100:
94:
92:
51:
27:
19:
1278:
1277:
1273:
1272:
1271:
1269:
1268:
1267:
1253:
1252:
1244:
1239:
1238:
1229:
1227:
1219:
1218:
1214:
1206:
1199:
1198:
1191:
1182:
1180:
1176:
1175:
1171:
1161:
1159:
1150:MacMahon, P. A.
1148:
1147:
1126:
1121:
1116:
1115:
1110:
1106:
1100:
1096:
1091:
1059:
1044:
1040:
1036:
1029:
1023:
1009:
1004:
1000:
993:
984:
977:
975:
971:
966:
957:
955:
951:
946:
937:
935:
931:
926:
914:
876:
855:
854:
822:
809:
796:
757:
756:
724:
703:
702:
670:
649:
648:
625:
624:
585:
546:
545:
526:
525:
507:
166:
128:
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
1276:
1274:
1266:
1265:
1255:
1254:
1251:
1250:
1243:
1242:External links
1240:
1237:
1236:
1212:
1209:. p. 184.
1189:
1169:
1123:
1122:
1120:
1117:
1114:
1113:
1104:
1093:
1092:
1090:
1087:
1058:
1055:
1042:
1038:
1034:
1033:For example, B
1027:
1022:
1019:
1007:
1002:
998:
997:For example, C
991:
983:
982:Contact system
980:
979:
978:
973:
969:
967:
960:
958:
953:
949:
947:
940:
938:
933:
929:
927:
920:
918:
913:
910:
906:
905:
894:
891:
888:
883:
879:
875:
872:
867:
864:
851:
840:
837:
834:
829:
825:
821:
816:
812:
808:
803:
799:
795:
792:
789:
786:
783:
780:
777:
774:
769:
766:
753:
742:
739:
736:
731:
727:
723:
720:
715:
712:
699:
688:
685:
682:
677:
673:
669:
666:
661:
658:
632:
609:
606:
603:
600:
597:
592:
588:
584:
581:
578:
575:
572:
569:
566:
563:
558:
555:
533:
506:
503:
500:
499:
496:
493:
490:
487:
484:
481:
478:
474:
473:
466:
459:
452:
445:
438:
435:
428:
421:
418:
410:
409:
406:
403:
400:
397:
394:
391:
388:
384:
383:
376:
369:
362:
355:
352:
345:
338:
335:
328:
320:
319:
316:
313:
310:
307:
304:
301:
298:
294:
293:
286:
279:
272:
269:
262:
255:
252:
245:
238:
230:
229:
226:
223:
220:
216:
215:
212:
209:
165:
162:
142:Percy MacMahon
126:
117:
116:
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
1275:
1264:
1261:
1260:
1258:
1249:
1246:
1245:
1241:
1226:
1222:
1216:
1213:
1205:
1204:
1196:
1194:
1190:
1179:
1173:
1170:
1157:
1156:
1151:
1145:
1143:
1141:
1139:
1137:
1135:
1133:
1131:
1129:
1125:
1118:
1108:
1105:
1098:
1095:
1088:
1086:
1084:
1083:
1077:
1075:
1071:
1066:
1062:
1056:
1054:
1052:
1046:
1031:
1020:
1018:
1016:
1011:
995:
989:
986:In his book,
981:
964:
959:
944:
939:
924:
919:
916:
911:
909:
889:
886:
881:
877:
870:
865:
862:
852:
835:
832:
827:
823:
819:
814:
810:
806:
801:
797:
790:
784:
781:
778:
772:
767:
764:
754:
737:
734:
729:
725:
718:
713:
710:
700:
683:
680:
675:
671:
664:
659:
656:
646:
645:
644:
630:
621:
604:
601:
598:
595:
590:
586:
579:
573:
570:
567:
561:
556:
553:
531:
522:
520:
516:
512:
504:
497:
494:
491:
488:
485:
482:
479:
476:
475:
471:
467:
464:
460:
457:
453:
450:
446:
443:
439:
433:
429:
426:
422:
416:
412:
411:
407:
404:
401:
398:
395:
392:
389:
386:
385:
381:
377:
374:
370:
367:
363:
360:
356:
350:
346:
343:
339:
333:
329:
326:
322:
321:
317:
314:
311:
308:
305:
302:
299:
296:
295:
291:
287:
284:
280:
277:
273:
267:
263:
260:
256:
250:
246:
243:
239:
236:
232:
231:
218:
217:
214:Three colors
208:Single color
207:
206:
200:
198:
197:supercomputer
193:
191:
187:
183:
175:
170:
163:
161:
159:
155:
151:
147:
143:
139:
135:
123:
113:
110:
102:
91:
88:
84:
81:
77:
74:
70:
67:
63:
60: –
59:
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
1228:. Retrieved
1224:
1215:
1202:
1181:. Retrieved
1172:
1160:. Retrieved
1154:
1107:
1097:
1080:
1078:
1067:
1063:
1060:
1047:
1032:
1024:
1012:
996:
985:
907:
622:
523:
514:
508:
194:
185:
179:
173:
158:tessellation
133:
132:
105:
96:
86:
79:
72:
65:
53:
41:Please help
36:verification
33:
1162:19 December
511:NP-complete
211:Two colors
195:In 1964, a
1230:2021-04-06
1183:2021-04-06
1119:References
1021:Boundaries
853:Heptagon:
701:Pentagon:
647:Triangle:
69:newspapers
1070:Wang Tile
1043:22,16,8,2
976:(Fig. 28)
956:(Fig. 28)
936:(Fig. 27)
871:⋅
807:−
791:⋅
773:⋅
755:Hexagon:
719:⋅
665:⋅
596:−
580:⋅
562:⋅
1257:Category
1152:(1921).
1102:2 again.
1074:TetraVex
1015:relative
988:MacMahon
912:Variants
519:dominoes
228:Doubles
225:Doubles
222:Triples
164:The game
99:May 2021
1263:Puzzles
1082:Trioker
1039:10,10,0
1008:1,2,2,2
934:10,10,0
437:
420:
354:
337:
271:
254:
156:. Such
146:squares
136:are an
83:scholar
1072:, and
1051:parity
1035:20,0,0
974:12,4,4
954:20,0,0
505:Theory
219:Quads
150:colors
127:20,0,0
85:
78:
71:
64:
56:
1207:(PDF)
1089:Notes
1028:a,b,c
999:1,1,1
992:a,b,c
930:1,1,1
498:1323
495:1332
489:2323
483:2333
477:3333
408:1232
405:1322
399:2233
393:2223
387:2222
318:1213
315:1132
309:1133
303:1113
297:1111
190:up to
90:JSTOR
76:books
1164:2023
492:1233
486:1313
480:1333
402:1223
396:1212
390:1222
312:1123
306:1122
300:1112
154:grid
62:news
1003:1,2
970:1,2
950:1,2
45:by
1259::
1223:.
1192:^
1127:^
1076:.
1017:.
620:.
1233:.
1186:.
1166:.
1026:B
1006:C
972:B
968:C
952:B
948:C
932:B
928:C
893:)
890:6
887:+
882:6
878:n
874:(
866:7
863:n
839:)
836:2
833:+
828:2
824:n
820:+
815:3
811:n
802:4
798:n
794:(
788:)
785:1
782:+
779:n
776:(
768:6
765:n
741:)
738:4
735:+
730:4
726:n
722:(
714:5
711:n
687:)
684:2
681:+
676:2
672:n
668:(
660:3
657:n
631:n
608:)
605:2
602:+
599:n
591:2
587:n
583:(
577:)
574:1
571:+
568:n
565:(
557:4
554:n
532:n
112:)
106:(
101:)
97:(
87:·
80:·
73:·
66:·
39:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
