187:
101:
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is a Greek root meaning twenty. On a dodecahedron with labeled vertices, there are 30 different ways that these vertices could be connected to each other to form a
Hamiltonian cycle. However, without the labels, all Hamiltonian cycles are symmetric to each other under rotations and reflections of the
153:
describes the shape of any possible solution, in a way that can be remembered by game players. A completed polygon must cut the twelve faces of the dodecahedron into two strips of six pentagons. As this strip passes through each of its four middle pentagons, in turn, it connects through two edges of
78:
Hamilton sold his work to a game manufacturing company, and it was marketed both in the UK and Europe, but it was too easy to become commercially successful. Only a small number of copies of it are known to survive in museums. Although
Hamilton was not the first to study Hamiltonian cycles, his work
173:
The game was too easy to play to achieve much popularity, although
Hamilton tried to counter this impression by giving an example of an academic colleague who failed to solve it. David Darling suggests that Hamilton may have made it much more difficult for himself than for others, by using his
316:
in
England was also studying Hamiltonian cycles on polyhedra. Hamilton visited Kirkman in 1861, and presented him with a copy of the icosian game. Despite this related work, some of which was much earlier, Hamiltonian cycles came to be named for Hamilton and for his work on the icosian game.
283:
Several versions of the game were sold in Europe. However, it was not a commercial success. Hamilton received only a ÂŁ25 licensing fee from Jaques and Son for his invention (present value ÂŁ3200). Few original copies of the game are known to survive, but one is kept in the library of the
277:
The
Icosian Game, invented by Sir William Rowan Hamilton, Royal Astronomer of Ireland; forming a new and highly amusing game for the drawing room, particularly interesting to students in mathematics of illustrating the principles of the Icosian
165:), with holes for numbered pegs to be placed at its vertices. The polygon found by game players was indicated by the consecutive numbering of the pegs. Another version was shaped as a "partially flattened dodecahedron", a roughly hemispherical
169:
with the pentagons of a dodecahedron spread on its curved surface and a handle attached to its flat base. The vertices had fixed pegs. A separate string, with a loop at one end, was wound through these pegs to indicate the polygon.
240:
The name of the icosian game comes from the fact that the icosahedron has twenty faces, the dodecahedron has twenty vertices, and any polygon through all the vertices of the dodecahedron has twenty edges.
154:
each pentagon that are not adjacent, making either a shallow left turn or a shallow right turn through the pentagon. In this way, the strip makes two left turns and then two right turns, or vice versa.
885:
466:
37:
147:.) In a two-player version of the game, one player starts by choosing five consecutive vertices along the polygon, and the other player must complete the polygon.
917:
1055:
Fernau, Henning; Haase, Carolina; Hoffmann, Stefan (2022), "The synchronization game on subclasses of automata", in
Fraigniaud, Pierre; Uno, Yushi (eds.),
261:
859:
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947:
336:. Puzzles like Hamilton's icosian game, based on finding Hamiltonian cycles in planar graphs, continue to be sold as smartphone apps.
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575:
749:
456:"Les jeux dans les collections du Conservatoire national des arts et métiers de Paris, 1 – le Jeu icosien (1859) (1re partie)"
83:
studied his game. Other puzzles based on
Hamiltonian cycles are sold as smartphone apps, and mathematicians continue to study
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1161:
1114:
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This company marketed
Hamilton's game beginning in 1859, in both its handheld solid and flat forms, under the lengthy titles
143:, passing exactly once through each vertex of the dodecahedron. A polygon visiting all vertices in this way is now called a
1057:
11th
International Conference on Fun with Algorithms, FUN 2022, May 30 to June 3, 2022, Island of Favignana, Sicily, Italy
909:
803:
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828:(May 1957), "Mathematical Games: About the remarkable similarity between the Icosian Game and the Tower of Hanoi",
1254:
202:
367:
256:
Both the icosian calculus and the icosian game were outlined by
Hamilton in a series of letters to his friend
1192:
300:
Although Hamilton invented the icosian game independently, he was not the first to study Hamiltonian cycles.
17:
790:, International Series in Operations Research & Management Science, New York: Springer, pp. 3–4,
1239:
210:
221:. The motivation for Hamilton was the problem of understanding the symmetries of the dodecahedron and
564:, Princeton Series in Applied Mathematics, vol. 17, Princeton University Press, pp. 18–20,
341:
71:. Hamilton's invention of the game came from his studies of symmetry, and from his invention of the
321:
80:
1037:
786:; Ejov, Vladimir; Filar, Jerzy A.; Nguyen, Giang T. (2012), "1.1: The graph that started it all",
1059:, LIPIcs, vol. 226, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, pp. 14:1–14:17,
706:
688:
607:
234:
905:
684:
308:, another puzzle based on Hamiltonian cycles, go back to the 9th century, both in India and in
206:
198:
190:
52:
914:
Report of the Twenty-Seventh Meeting of the British Association for the Advancement of Science
864:
Resources for Teaching Discrete Mathematics: Classroom Projects, History Modules, and Articles
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8:
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695:, vol. 3, Hodges, Figgis, & Co., and Longman's, Green, & Co., pp. 55–56
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1148:, Oberwolfach Seminars, vol. 44, Basel: Birkhäuser / Springer, pp. 75–84,
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709:(June 2016), "Sir William Rowan Hamilton: Life, achievements, stature in physics",
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on this game became the origin of the name of Hamiltonian cycles. Several works of
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in late 1856. Hamilton then exhibited the game at the 1857 Dublin meeting of the
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that have the same symmetries as each other. For this purpose he also invented
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Bulletin de la Société archéologique, historique et artistique le Vieux papier
157:
One version of the game took the form of a flat wooden board inscribed with a
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855:
498:
Turner, Gerard L'E. (October 1987), "Scientific toys", Presidential address,
1087:
1083:
349:
1140:
Hefetz, Dan; Krivelevich, Michael; Stojaković, Miloš; Szabó, Tibor (2014),
1011:
993:
860:"Early Writings on Graph Theory: Hamiltonian Circuits and The Icosian Game"
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111:
60:
222:
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218:
75:, a mathematical system describing the symmetries of the dodecahedron.
264:. At the suggestion of Graves, Hamilton sold its publishing rights to
100:
1220:
36:
Modern reconstruction of Hamilton's icosian game, on display at the
436:(in French), vol. 2, Paris: Gauthier-Villars, pp. 201–227
119:
32:
1044:(in German), vol. 2, Leipzig: B. G. Teubner, pp. 196–210
136:
64:
1139:
320:
The icosian game itself has been the topic of multiple works in
38:
Institute of Mathematics and Statistics, University of SĂŁo Paulo
243:
161:
with the same combinatorial structure as the dodecahedron (a
288:
in Dublin, and another is included in the collection of the
67:
using edges of the dodecahedron that passes through all its
166:
370:, a puzzle of finding a cycle through all edges of a graph
866:, Mathematical Association of America, pp. 217–224,
539:
273:
The Travellers Dodecahedron, or a voyage around the world
970:
Across the Board: The Mathematics of Chessboard Problems
968:
Watkins, John J. (2004), "Chapter 2: Knight's Tours",
958:; includes color photographs of both original versions
1209:
782:
558:
The Traveling Salesman Problem: A Computational Study
268:, a London-based toy and game manufacturing company.
364:, another game played on the graph of a dodecahedron
1054:
642:"Hamilton's icosian calculus and his icosian game"
262:British Association for the Advancement of Science
197:At the time of his invention of the icosian game,
135:The game's object is to find a three-dimensional
1231:
999:The Bulletin of the London Mathematical Society
340:based on Hamiltonian cycles were introduced to
324:by well-known authors on the subject including
1226:, showing the 30 solutions on the planar board
1202:, including an interactive solver for finding
972:, Princeton University Press, pp. 25–38,
501:The British Journal for the History of Science
1072:
386:
384:
352:, and continue to be studied in mathematics.
292:in Paris, both in the flat form of the game.
1078:
788:Hamiltonian Cycle Problem and Markov Chains
237:which he used to compute these symmetries.
174:theoretical methods to solve it instead of
850:
848:
820:
818:
816:
814:
390:
381:
290:Conservatoire national des arts et métiers
1064:
1038:"XVII: Das Hamiltonische Dodekaederspiel"
747:(1995), "The icosian calculus of today",
598:
596:
594:
430:"Septième récréation – le jeu d'Hamilton"
310:mathematics in the medieval Islamic world
213:, and was already famous for his work on
1187:
996:(1981), "T. P. Kirkman, mathematician",
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51:invented in 1856 by Irish mathematician
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27:Game of finding cycles on a dodecahedron
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1042:Mathematische Unterhaltungen und Spiele
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525:; see p. 395 and photo, Fig. 13, p. 397
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750:Proceedings of the Royal Irish Academy
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646:Humanistic Mathematics Network Journal
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312:. At about the same time as Hamilton,
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1142:"Chapter 6: The Hamiltonicity game"
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24:
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916:, London: John Murray, p. 3,
693:Life of Sir William Rowan Hamilton
528:
193:, the inventor of the icosian game
25:
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118:
99:
950:from the original on 2024-01-21
920:from the original on 2024-04-25
888:from the original on 2024-04-25
664:from the original on 2024-03-11
618:from the original on 2024-04-25
581:from the original on 2021-05-06
472:from the original on 2024-04-26
1095:Annals of Discrete Mathematics
400:Graph Theory with Applications
203:Andrews Professor of Astronomy
13:
1:
1107:10.1016/S0167-5060(08)70335-2
834:, vol. 196, no. 5,
454:Boutin, Michel (April 2018),
402:, North Holland, p. 53,
374:
181:
87:based on Hamiltonian cycles.
1250:Hamiltonian paths and cycles
938:Dalgety, James (July 2002),
872:10.5948/upo9780883859742.028
465:(in French) (428): 433–441,
90:
7:
1154:10.1007/978-3-0348-0825-5_6
862:, in Hopkins, Brian (ed.),
368:Seven Bridges of Königsberg
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211:Royal Astronomer of Ireland
10:
1271:
1066:10.4230/LIPICS.FUN.2022.14
251:
1193:"Hamilton's Icosian Game"
1088:"Biased positional games"
910:"On the icosian calculus"
796:10.1007/978-1-4614-3232-6
723:10.1007/s12045-016-0356-y
658:10.5642/hmnj.200101.24.14
640:Sowell, Katye O. (2001),
514:10.1017/s0007087400024195
434:Récréations mathématiques
342:combinatorial game theory
295:
139:made from the edges of a
322:recreational mathematics
81:recreational mathematics
55:. It involves finding a
685:Graves, Robert Perceval
612:Encyclopedia of Science
235:non-commutative algebra
1255:William Rowan Hamilton
927:– via Hathitrust
207:Trinity College Dublin
199:William Rowan Hamilton
194:
191:William Rowan Hamilton
128:view of the same cycle
53:William Rowan Hamilton
40:
275:, and (respectively)
217:and his invention of
215:Hamiltonian mechanics
189:
35:
1012:10.1112/blms/13.2.97
856:Barnett, Janet Heine
141:regular dodecahedron
831:Scientific American
541:Applegate, David L.
344:in a 1978 paper by
338:Maker-Breaker games
286:Royal Irish Academy
85:combinatorial games
1240:Mathematical games
1212:Weisstein, Eric W.
1198:The New York Times
1191:(6 October 2014),
940:"The Icosian Game"
689:"The Icosian Game"
652:(24), Article 14,
195:
41:
1206:on a dodecahedron
1204:Hamiltonian paths
1163:978-3-0348-0824-8
1116:978-0-7204-1043-3
979:978-0-691-15498-5
944:The Puzzle Museum
881:978-0-88385-974-2
145:Hamiltonian cycle
108:Hamiltonian cycle
57:Hamiltonian cycle
49:mathematical game
16:(Redirected from
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73:icosian calculus
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176:trial and error
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1216:"Icosian game"
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717:(6): 493–510,
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608:"Icosian game"
604:Darling, David
590:
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549:Chvátal, Vašek
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508:(4): 377–398,
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426:Lucas, Édouard
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346:Václav Chvátal
334:Martin Gardner
330:Wilhelm Ahrens
314:Thomas Kirkman
302:Knight's tours
297:
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266:Jaques and Son
258:John T. Graves
253:
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248:dodecahedron.
233:, a system of
227:dual polyhedra
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1245:Graph theory
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994:Biggs, N. L.
988:
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952:, retrieved
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922:, retrieved
913:
900:
890:, retrieved
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666:, retrieved
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583:, retrieved
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474:, retrieved
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392:Bondy, J. A.
319:
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159:planar graph
156:
149:
134:
112:dodecahedron
77:
61:dodecahedron
45:icosian game
44:
42:
29:
18:Icosian Game
1101:: 221–229,
1080:Chvátal, V.
707:Mukunda, N.
306:chessboards
223:icosahedron
219:quaternions
1234:Categories
954:2024-04-25
924:2024-04-25
892:2024-04-25
668:2024-04-25
622:2024-04-24
585:2024-04-25
476:2024-04-25
375:References
350:Paul Erdős
182:Background
1221:MathWorld
1084:Erdős, P.
757:: 23–34,
711:Resonance
91:Game play
1086:(1978),
1036:(1918),
948:archived
918:archived
908:(1858),
886:archived
858:(2009),
840:24940862
763:20490184
687:(1889),
662:archived
616:archived
576:archived
555:(2011),
467:archived
428:(1883),
398:(1976),
356:See also
278:Calculus
201:was the
69:vertices
1172:3524719
1125:0500701
1020:0608093
771:1649815
522:4026415
252:History
137:polygon
65:polygon
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802:
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406:
332:, and
296:Legacy
225:, two
126:Planar
1091:(PDF)
836:JSTOR
759:JSTOR
579:(PDF)
562:(PDF)
518:JSTOR
470:(PDF)
459:(PDF)
244:Icosa
110:on a
59:on a
47:is a
1158:ISBN
1111:ISBN
974:ISBN
876:ISBN
800:ISBN
566:ISBN
404:ISBN
348:and
209:and
167:dome
63:, a
43:The
1150:doi
1103:doi
1061:doi
1008:doi
868:doi
792:doi
719:doi
654:doi
510:doi
304:on
205:at
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