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The
Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems: Number Theory, Algebra, Geometry, Probability, Topology, Game Theory, Infinity, and Other Topics of Recreational Mathematics.
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means that the second player can never win. All that is left to study is to determine whether the first player can win, on what board sizes he may do so, and in how many moves it will take.
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335:, Encyclopedia of Mathematics and its Applications, vol. 114, Cambridge: Cambridge University Press, pp. 60–64,
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Harary's generalization does not include tic-tac-toe itself, as diagonal constructions are not considered a win.
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be the smallest number of moves in which the first player can force a win, assuming perfect play by both sides.
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on a square grid of varying size, rather than being limited to "in a row" constructions. It was devised by
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261:(with a possible exception of the N-hexomino, which is still currently unsolved, may have
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How to find winning
Strategy for 4 celled animals of Harary's generalized tic tac toe
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be the smallest size square board on which the first player can win, and let
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1st ed. New York: W. W. Norton & Company, 2001. 286-311.
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in March 1977, and is a broader definition than that of an
41:, defining the game as a race to complete a particular
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254:Z-pentomino: The first player cannot win
240:X-pentomino: The first player cannot win
237:W-pentomino: The first player cannot win
234:V-pentomino: The first player cannot win
231:U-pentomino: The first player cannot win
228:T-pentomino: The first player cannot win
225:P-pentomino: The first player cannot win
200:I-pentomino: The first player cannot win
168:O-tetromino: The first player cannot win
332:Combinatorial Games: Tic-Tac-Toe Theory
329:(2008), "Harary's Animal Tic-Tac-Toe",
305:QBF Encoding of Generalized Tic-Tac-Toe
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276:and above: The first player cannot win
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18:Harary's generalized tictactoe
269:= 13): The first player cannot win
59:Like many other two-player games,
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523:Harary's generalized tic-tac-toe
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37:is a generalization of the game
31:Harary's generalized tic-tac-toe
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197:: The first player cannot win
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533:Strategy-stealing argument
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341:10.1017/CBO9780511735202
543:Paper-and-pencil game
528:Hales–Jewett theorem
464:Ultimate tic-tac-toe
449:Quantum tic-tac-toe
642:Mathematical games
586:Three men's morris
35:animal tic-tac-toe
629:
628:
559:Nine men's morris
61:strategy stealing
27:Mathematical game
16:(Redirected from
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518:Kaplansky's game
487:Related concepts
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469:Wild tic-tac-toe
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439:Order and Chaos
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552:Similar games
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243:Y-pentomino:
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214:N-pentomino:
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203:L-pentomino:
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182:Z-tetromino:
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171:T-tetromino:
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157:L-tetromino:
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72:Square boards
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601:Connect Four
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568:Tic-Stac-Toe
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327:Beck, József
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47:Frank Harary
34:
30:
29:
647:Tic-tac-toe
616:Toss Across
538:Futile game
427:Treblecross
396:Tic-tac-toe
294:Tic-tac-toe
274:heptominoes
131:V-tromino:
39:tic-tac-toe
636:Categories
591:Nine Holes
564:Score Four
281:References
259:hexominoes
51:m,n,k-game
265:= 15 and
195:pentomino
144:tetromino
43:polyomino
606:Connect6
404:Variants
89:monomino
621:Pentago
574:Gobblet
422:Notakto
349:2402857
118:tromino
67:Results
580:Quarto
417:Gomoku
347:
103:domino
505:-game
454:Renju
444:Pente
247:= 7,
218:= 6,
207:= 7,
186:= 3,
175:= 5,
161:= 4,
150:= 7,
135:= 3,
124:= 4,
109:= 2,
95:= 1,
596:Achi
513:game
272:All
257:All
211:= 10
76:Let
611:OXO
459:SOS
337:doi
251:= 9
222:= 6
190:= 5
179:= 4
165:= 4
154:= 8
139:= 3
128:= 3
113:= 2
99:= 1
33:or
638::
357:.
345:MR
343:,
193:F-
146::
142:I-
120::
116:I-
105::
91::
53:.
511:n
503:k
501:,
499:n
497:,
495:m
388:e
381:t
374:v
339::
267:m
263:b
249:m
245:b
220:m
216:b
209:m
205:b
188:m
184:b
177:m
173:b
163:m
159:b
152:m
148:b
137:m
133:b
126:m
122:b
111:m
107:b
97:m
93:b
82:m
78:b
20:)
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