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233:), there will always be a winning strategy for player X. There will never be a winning strategy for player O because of the
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Given a width of tic-tac-toe board, what is the smallest dimension such that X is guaranteed a winning strategy?
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cannot be won if both players are playing well as an opponent's piece will block the one-dimensional line.
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Tichy, Robert F.; Schlickewei, Hans Peter; Schmidt, Klaus D. (10 July 2008).
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spaces are filled then the game is a draw. Tic-tac-toe is the game where
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games are trivially won by the first player as there is only one space (
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of their symbol (X or O) they win the game. However, if all
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16:
Generalization of the game Tic-tac-toe to higher dimensions
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207:{\displaystyle {\frac {\left(n+2\right)^{d}-n^{d}}{2}}}
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with 2 players. If one player creates a line of length
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402:"A Scientific Study: k-dimensional Tic-Tac-Toe"
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43:to higher dimensions. It is a game played on a
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428:The Theory of Gambling and Statistical Logic
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770:
491:
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425:Epstein, Richard A. (28 December 2012).
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306:Combinatorial Games: Tic-Tac-Toe Theory
130:(more unsolved problems in mathematics)
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253: – Degenerate tic-tac toe variant
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808:Unsolved problems in mathematics
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627:Harary's generalized tic-tac-toe
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399:
364:Golomb, Solomon; Hales, Alfred.
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121:Unsolved problem in mathematics
456:Higher-Dimensional Tic-Tac-Toe
418:
323:
309:. Cambridge University Press.
303:Beck, JĂłzsef (20 March 2008).
114:
1:
257:
756:. You can help Knowledge by
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637:Strategy-stealing argument
235:Strategy-stealing argument
39:of the combinatorial game
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590:
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366:"Hypercube Tic-Tac-Toe"
752:-related article is a
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647:Paper-and-pencil game
209:
149:There are a total of
632:Hales–Jewett theorem
568:Ultimate tic-tac-toe
231:Hales-Jewett theorem
225:, at some dimension
153:
553:Quantum tic-tac-toe
460:PBS Infinite Series
237:since an n game is
214:winning lines in a
690:Three men's morris
431:. Academic Press.
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144:combinatorial game
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663:Nine men's morris
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66:equals 2 (3, 2).
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622:Kaplansky's game
591:Related concepts
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573:Wild tic-tac-toe
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378:on 29 April 2016
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371:. Archived from
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543:Order and Chaos
538:Number Scrabble
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229:(thanks to the
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516:3D tic-tac-toe
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450:External links
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275:"Mathllaneous"
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221:For any width
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37:generalization
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656:Similar games
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400:Shih, Davis.
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62:equals 3 and
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758:expanding it
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705:Connect Four
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672:Tic-Stac-Toe
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408:. Retrieved
380:. Retrieved
373:the original
336:. Springer.
332:
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281:. Retrieved
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803:Tic-tac-toe
720:Toss Across
642:Futile game
531:Treblecross
500:Tic-tac-toe
410:16 December
382:16 December
283:16 December
251:Treblecross
115:Game theory
41:tic-tac-toe
813:Game stubs
797:Categories
695:Nine Holes
668:Score Four
258:References
139:game is a
80:> 0, 0)
74:game. The
458:from the
239:symmetric
186:−
141:symmetric
48:hypercube
710:Connect6
508:Variants
245:See also
725:Pentago
678:Gobblet
526:Notakto
464:YouTube
70:is the
35:) is a
684:Quarto
521:Gomoku
435:
340:
313:
218:game.
109:> 1
84:(1, 1)
72:(4, 3)
748:This
609:-game
558:Renju
548:Pente
405:(PDF)
376:(PDF)
369:(PDF)
278:(PDF)
95:1 = 1
68:Qubic
754:stub
750:game
700:Achi
617:game
433:ISBN
412:2016
384:2016
338:ISBN
311:ISBN
285:2016
104:and
93:and
33:game
28:(or
26:game
715:OXO
563:SOS
462:on
135:An
102:= 1
91:= 1
82:or
799::
392:^
352:^
293:^
266:^
241:.
146:.
21:A
785:e
778:t
771:v
760:.
615:n
607:k
605:,
603:n
601:,
599:m
492:e
485:t
478:v
441:.
414:.
386:.
346:.
319:.
287:.
227:d
223:n
216:n
200:2
194:d
190:n
181:d
176:)
172:2
169:+
166:n
162:(
137:n
123::
107:n
100:d
89:n
78:n
76:(
64:d
60:n
56:n
52:n
45:n
31:n
24:n
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.