349:, who played Hex at Princeton in the 1940s, so that Nash may have subconsciously picked up the idea. Hein wrote to Gardner in 1957 expressing doubt that Nash discovered Hex independently. Gardner was unable to independently verify or refute Nash's claim. Gardner privately wrote to Hein: "I discussed it with the editor, and we decided that the charitable thing to do was to give Nash the benefit of the doubt. ... The fact that you invented the game before anyone else is undisputed. Any number of people can come along later and say that they thought of the same thing at some later date, but this means little and nobody really cares." In a later letter to Hein, Gardner also wrote: "Just between you and me, and off the record, I think you hit the nail on the head when you referred to a 'flash of a suggestion' which came to Mr. Nash from a Danish source, and which he later forgot about. It seems the most likely explanation."
719:
boards in 2002 and the center opening on 9×9 boards in 2003. In 2009, Philip
Henderson, Broderick Arneson and Ryan B. Hayward completed the analysis of the 8×8 board with a computer search, solving all the possible openings. In 2013, Jakub Pawlewicz and Ryan B. Hayward solved all openings for 9×9 boards, and one (the most-central) opening move on the 10×10 board. Since Gardner first postulated in his column in Scientific American in 1957, albeit speciously, that any first play on the short diagonal is a winning play, for all solved game boards up to n=9, that has indeed been the case. In addition, for all boards except n=2 and n=4, there have been numerous additional winning first moves; the number of winning first moves generally is ≥ n²/2.
576:
the same color and two empty spaces, where the two stones do not touch. If the opponent plays in either space, the player plays in the other, creating a contiguous chain. There are also safely connected patterns which connect stones to edges. There are many more safely connected patterns, some quite complex, built up of simpler ones like those shown. Patterns and paths can be disrupted by the opponent before they are complete, so the configuration of the board during an actual game often looks like a patchwork rather than something planned or designed.
262:
36:
525:
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mentioned it as one of his design criteria for Hex in the original
Politiken article. Hein also stated this fact as "a barrier for your opponent is a connection for you". John Nash wrote up a proof of this fact around 1949, but apparently did not publish the proof. Its first exposition appears in an in-house technical report in 1952, in which Nash states that "connection and blocking the opponent are equivalent acts". A more rigorous proof was published by
358:
157:. Each player is assigned a pair of opposite sides of the board, which they must try to connect by alternately placing a stone of their color onto any empty hex. Once placed, the stones are never moved or removed. A player wins when they successfully connect their sides together through a chain of adjacent stones. Draws are impossible in Hex due to the
783:
are further apart can win regardless of who plays first. On boards with equal dimensions, the first player can win on a board with an even number of cells per side, and the second player can win on a board with an odd number. On boards with an even number, one of the first player's winning moves is always to place a stone in the acute corner.
345:, Nash's fellow players called the game either Nash or John, with the latter name referring to the fact that the game could be played on hexagonal bathroom tiles. Nash insisted that he discovered the game independently of Hein, but there is some doubt about this, as it is known that there were Danish people, including
852:
Dark Hex (also known as
Phantom Hex) is an imperfect information version of Hex. The players are not exposed to each other's stones at any point in the game unless they discover them first. The game is played in the presence of an umpire where each player first verifies the move if its a collision or
621:
An informal proof of the no-draw property of Hex can be sketched as follows: consider the connected component of one of the red edges. This component either includes the opposite red edge, in which case Red has a connection, or else it does not, in which case the blue stones along the boundary of the
575:
A "safely connected" pattern is composed of stones of the player's color and open spaces which can be joined into a chain, an unbroken sequence of edge-wise adjacent stones, no matter how the opponent plays. One of the simplest such patterns is the bridge, which consists of a diamond of two stones of
563:
From the proof of a winning strategy for the first player, it is known that the Hex board must have a complex type of connectivity which has never been solved. Play consists of creating small patterns which have a simpler type of connectivity called "safely connected", and joining them into sequences
277:
The players take turns placing a stone of their color on a single cell on the board. The most common convention is for Red or Black to go first. Once placed, stones are not moved, replaced, or removed from the board. Each player's goal is to form a connected path of their own stones linking their two
667:
The first player can now adopt the following strategy. They make an arbitrary move. Thereafter they play the winning second player strategy assumed above. If in playing this strategy, they are required to play on the cell where an arbitrary move was made, they make another arbitrary move. In this
273:
grid of hexagons, typically of size 11×11, although other sizes are also possible. Each player has an allocated color, conventionally red and blue, or black and white. Each player is also assigned two opposite board edges. The hexagons on each of the four corners belong to both adjacent board edges.
1352:
Cazenave, Tristan; Chen, Yen-Chi; Chen, Guan-Wei; Chen, Shi-Yu; Chiu, Xian-Dong; Dehos, Julien; Elsa, Maria; Gong, Qucheng; Hu, Hengyuan; Khalidov, Vasil; Li, Cheng-Ling; Lin, Hsin-I; Lin, Yu-Jin; Martinet, Xavier; Mella, Vegard; Rapin, Jeremy; Roziere, Baptiste; Synnaeve, Gabriel; Teytaud, Fabien;
642:, the first player has a theoretical winning strategy. This fact was mentioned by Hein in his notes for a lecture he gave in 1943: "in contrast to most other games, it can be proved that the first player in theory always can win, that is, if she could see to the end of all possible lines of play".
872:
tournaments reported from Brazil, Czech
Republic, Denmark, France, Germany, Italy, Netherlands, Norway, Poland, Portugal, Spain, UK and the US. One of the largest Hex competitions is organized by the International Committee of Mathematical Games in Paris, France, which is annually held since 2013.
782:
variant of Hex is called "Rex", in which each player tries to force their opponent to make a chain. Rex is slower than Hex since, on any empty board with equal dimensions, the losing player can delay a loss until the entire board is full. On boards with unequal dimensions, the player whose sides
579:
There are weaker types of connectivity than "safely connected" which exist between stones or between safely connected patterns which have multiple spaces between them. The middle part of the game consists of creating a network of such weakly connected stones and patterns which hopefully will allow
747:
The game may be played on a rectangular grid like a chess, checker or go board, by considering that spaces (intersections in the case of go) are connected in one diagonal direction but not the other. The game may be played with paper and pencil on a rectangular array of dots or graph paper in the
718:
In 2002, Jing Yang, Simon Liao and Mirek Pawlak found an explicit winning strategy for the first player on Hex boards of size 7×7 using a decomposition method with a set of reusable local patterns. They extended the method to weakly solve the center pair of topologically congruent openings on 8×8
597:
It is not difficult to convince oneself by exposition, that hex cannot end in a draw, referred to as the "hex theorem". I.e., no matter how the board is filled with stones, there will always be one and only one player who has connected their edges. This fact was known to Piet Hein in 1942, who
706:
to Hex. This result means that there is no efficient (polynomial time in board size) algorithm to solve an arbitrary Hex position unless there is an efficient algorithm for all PSPACE problems, which is widely believed not to be the case. However, it doesn't rule out the possibility of a simple
418:
constructed an analog Hex playing machine, which was essentially a resistance network with resistors for edges and light bulbs for vertices. The move to be made corresponded to a certain specified saddle point in the network. The machine played a reasonably good game of Hex. Later, researchers
645:
All known proofs of this fact are non-constructive, i.e., the proof gives no indication of what the actual winning strategy is. Here is a condensed version of a proof that is attributed to John Nash c. 1949. The proof works for a number of games including Hex, and has come to be called the
583:
Success at Hex requires a particular ability to visualize synthesis of complex patterns in a heuristic way, and estimating whether such patterns are 'strongly enough' connected to enable an eventual win. The skill is somewhat similar to the visualization of patterns, sequencing of moves, and
813:
The game of Y is Hex played on a triangular grid of hexagons; the object is for either player to connect all three sides of the triangle. Y is a generalization of Hex to the extent that any position on a Hex board can be represented as an equivalent position on a larger Y board.
564:
that form a "path". Eventually, one of the players will succeed in forming a safely connected path of stones and spaces between their sides of the board and win. The final stage of the game, if necessary, consists of filling in the empty spaces in the path.
480:
Until 2019, humans remained better than computers at least on big boards such as 19x19, but on Oct 30, 2019 the program Mootwo won against the human player with the best Elo rank on LittleGolem, also winner of various tournaments (the game is available
799:. In order to play a "move", contestants had to answer a question correctly. The board had 5 alternating columns of 4 hexagons; the solo player could connect top-to-bottom in 4 moves, while the team of two could connect left-to-right in 5 moves.
622:
connected component form a winning path for Blue. The concept of a connected component is well-defined because in a hexagonal grid, two cells can only meet in an edge or not at all; it is not possible for cells to overlap in a single point.
365:
Initially in 1942, Hein distributed the game, which was then called
Polygon, in the form of 50-sheet game pads. Each sheet contained an empty 11×11 empty board that could be played on with pencils or pens. In 1952,
460:
Starting about 2006, the field of computer Hex came to be dominated by Monte Carlo tree search methods borrowed from successful computer implementations of Go. These replaced earlier implementations that combined
509:
and growing architectures (the program can learn on a small board, and then extrapolate on a big board, as opposed to justified popular claims about earlier artificial intelligence methods such as the original
288:
When it is clear to both players who will win the game, it is customary, but not required, for the losing player to resign. In practice, most games of Hex end with one of the players resigning.
285:(also called the pie rule) is normally used. This rule allows the second player to choose whether to switch positions with the first player after the first player makes the first move.
427:
It was known to Hein in 1942 that Hex cannot end in a draw; in fact, one of his design criteria for the game was that "exactly one of the two players can connect their two sides".
1947:
671:
This extra piece cannot interfere with the first player's imitation of the winning strategy, for an extra piece is never a disadvantage. Therefore, the first player can win.
674:
Because we have now contradicted our assumption that there is a winning strategy for the second player, we conclude that there is no winning strategy for the second player.
571:
A "bridge" (A ↔ C) is a simple example of a safely connected pattern. It consists of two stones of the same color (A and C), and a pair of open spaces (B and D).
944:
If the board has been completely filled, then one player must have won already, and it must be the first player since they have been playing a winning strategy.
828:
Havannah is game based on Hex. It differs from Hex in that it is played on a hexagonal grid of hexagons and a win is achieved by forming one of three patterns.
402:-inch (140 mm × 220 mm) 50-sheet pad of ruled Hex grids. Hex is currently published by Nestorgames in a 11×11 size, a 14×14, and a 19×19 size.
710:
In 11×11 Hex, the state space complexity is approximately 2.4×10; versus 4.6×10 for chess. The game tree complexity is approximately 10 versus 10 for chess.
1694:
374:
also sold a version under the "Con-tac-tix" name in 1968. Hex was also issued as one of the games in the 1974 3M Paper Games Series; the game contained a
1707:
H. J. van den Herik; J. W. H. M. Uiterwijk; J. van
Rijswijck (2002). "Games solved: Now and in the future". Artificial Intelligence. 134 (1–2): 277–311.
707:
winning strategy for the initial position (on boards of arbitrary size), or a simple winning strategy for all positions on a board of a particular size.
654:
Since Hex is a finite two-player game with perfect information either the first or second player has a winning strategy, or both can force a draw by
164:
Despite the simplicity of its rules, the game has deep strategy and sharp tactics. It also has profound mathematical underpinnings related to the
1489:
437:
1327:
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1790:
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attempting to solve the game and develop Hex-playing computer algorithms emulated
Shannon's network to create strong computer players.
1761:
1353:
Teytaud, Olivier; Ye, Shi-Cheng; Ye, Yi-Jun; Yen, Shi-Jim; Zagoruyko, Sergey (27 January 2020). "Polygames: Improved Zero
Learning".
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Popular dimensions other than the standard 11×11 are 13×13 and 19×19 as a result of the game's relationship to the older game of
2027:
1676:
444:, so a determinate winning strategy like that for the Shannon switching game on a regular rectangular grid was unavailable.
694:
proved that determining whether a position in a game of generalized Hex played on arbitrary graphs is a winning position is
580:
the player, by filling in the weak links, to construct just one safely connected path between sides as the game progresses.
1896:
1861:
1836:
1640:
Sanjeev Arora, Boaz Barak, "Computational
Complexity: A Modern Approach". Cambridge University Press, 2009. Section 4.3
1514:
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Since draws are impossible (see above), we can conclude that either the first or second player has a winning strategy.
2162:
2090:
2013:
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200:; they are no longer in production. Hex can also be played with paper and pencil on hexagonally ruled graph paper.
2147:
2115:
504:
433:
In 1952 John Nash wrote up an existence proof that on symmetrical boards, the first player has a winning strategy.
2172:
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loop. Like in Hex, there are no ties, and there is no position in which both players have a winning connection.
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1975:
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450:
In 2002, the first explicit winning strategy (a reduction-type strategy) on a 7×7 board was described.
1486:
1485:
Nash, John (Feb. 1952). Rand Corp. technical report D-1164: Some Games and
Machines for Playing Them.
2157:
2152:
342:
20:
1989:
2052:
469:. On the subject of early Computer Hex, notable early implementations include Dolphin Microware's
149:
board, although 13×13 and 19×19 boards are also popular. The board is composed of hexagons called
1101:
762:
703:
895:
457:
search computer algorithms, Hex boards up to size 9×9 (as of 2016) have been completely solved.
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313:. Although Hein later renamed it to Con-tac-tix, it became known in Denmark under the name
310:
139:
8:
1049:
209:
482:
1623:
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1375:
1354:
1256:
1193:
887:
841:
485:). This program was based on Polygames (an open-source project, initially developed by
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223:
173:
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1374:
Marcus, Gary (17 January 2018). "Innateness, AlphaZero, and Artificial Intelligence".
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2009:
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It was also known to Hein that the first player has a theoretical winning strategy.
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1921:
1627:
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1539:
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610:
published a proof that the determinacy of Hex is equivalent to the two-dimensional
1459:
Hayward, Ryan B.; Rijswijck, van, Jack (6 October 2006). "Hex and combinatorics".
317:
due to an article by Hein in the 26 December 1942 edition of the Danish newspaper
2065:
1765:
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1493:
1328:"Open-sourcing Polygames, a new framework for training AI bots through self-play"
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Other connection games with similar objectives but different structures include
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not. Based on the continuation of this point the game has different versions.
735:. Both of these bear some degree of similarity to the ancient Chinese game of
265:
The end of a game of Hex on a standard 11×11 board. Here, White wins the game.
2131:
2100:
1606:
Stefan Reisch (1981). "Hex ist PSPACE-vollständig (Hex is PSPACE-complete)".
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691:
107:
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way they play the winning strategy with one extra piece always on the board.
919:
793:
Hex had an incarnation as the question board from the television game show
321:, the first published description of the game, in which he used that name.
2086:
1583:
1566:
499:
boardsize invariance thanks to fully convolutional neural networks (as in
1768:, P. Henderson, B. Arneson, and R. Hayward, Proc. IJCAI-09 505-510 (2009)
687:
474:
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marketed a version of the game under the name "Hex", and the name stuck.
261:
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1619:
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246:
231:
130:
in which players attempt to connect opposite sides of a rhombus-shaped
127:
57:
1530:
David Gale (1979). "The Game of Hex and Brouwer Fixed-Point Theorem".
698:. A strengthening of this result was proved by Reisch by reducing the
278:
board edges. The player who complete such a connection wins the game.
35:
19:
This article is about the abstract strategy game. For other uses, see
2109:
1017:
The Scientific American Book of Mathematical Puzzles & Diversions
631:
494:
346:
282:
1748:
Unpublished white papers, formerly @ www.ee.umanitoba.com/~jingyang/
1543:
779:
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Consequently, there must be a winning strategy for the first player.
1487:
https://www.rand.org/content/dam/rand/pubs/documents/2015/D1164.pdf
1380:
1359:
1220:
Lehman, Alfred (1964). "A Solution of the Shannon Switching Game".
808:
770:(one of the game's inventors) advocated 14×14 as the optimal size.
169:
158:
1730:
On a decomposition method for finding winning strategy in Hex game
536: with: general strategy and typical endgames. You can help by
1961:
1210:
Anshelevich, V. (2002). A Hierarchical Approach to Computer Hex.
270:
185:
146:
131:
2058:
2055:
Hex strategy for beginners by Matthew Seymour and Eric Silverman
1720:. Ph.D. Thesis, University of Limburg, pdf, 6.3.9 Chess pp. 171
1567:"A Combinatorial Problem Which is Complete in Polynomial Space"
447:
In 1981, the Stefan Reisch showed that Hex is PSPACE-complete.
329:
The game was rediscovered in 1948 or 1949 by the mathematician
184:
on December 26, 1942. It was later marketed as a board game in
2089:
gathering theoretical work on Hex (moved - top level pages at
357:
1814:
Gardner, Martin, Scientific American, July, 1957, pgs 145-151
1228:(4). Society for Industrial and Applied Mathematics: 687–725.
732:
625:
618:-player variants proves the fixed-point theorem in general.
500:
302:
1718:
Searching for Solutions in Games and Artificial Intelligence
664:
Let us assume that the second player has a winning strategy.
40:
11×11 Hex gameboard showing a winning configuration for Blue
2075:
2071:
2022:, Hayward R. with Toft B.(2019), CRC Press Boca Raton, FL.
742:
713:
1458:
1239:
Reisch, Stefan (1981). "Hex ist PSPACE-vollständig".
1948:"Dark Hex: A Large Scale Imperfect Information Game"
1538:(10). Mathematical Association of America: 818–827.
1351:
1130:. Boca Raton, Florida: CRC Press. pp. 127–138.
281:
To compensate for the first player's advantage, the
241:Hex is a special case of the "node" version of the
2118:on A4 or A3 paper, for use with standard Go stones
1014:
138:in 1942 and later rediscovered and popularized by
2082:University of Alberta Computer Hex Research Group
1044:
1042:
748:same way by using two different colored pencils.
614:, and that the determinacy of higher-dimensional
2129:
2123:https://sites.google.com/view/cavegames-hex/home
2008:, Browne C.(2000), A.K. Peters Ltd. Natick, MA.
1777:
1693:: CS1 maint: bot: original URL status unknown (
877:. During this competition the pie rule is used.
840:, where the players have the goal of creating a
2049:Interactive tactical puzzles by Matthew Seymour
1856:. Boca Raton, Florida: CRC Press. p. 154.
1831:. Boca Raton, Florida: CRC Press. p. 175.
1267:
1178:Proceedings of the Institute of Radio Engineers
1079:. Boca Raton, Florida: CRC Press. p. 156.
984:
982:
980:
1739:, Jing Yang, Simon Liao and Mirek Pawlak, 2002
1654:. Natick, MA: A.K. Peters, Ltd. pp. 5–6.
1509:. Boca Raton, Florida: CRC Press. p. 99.
1176:Shannon, C. (1953). "Computers and Automata".
1039:
978:
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962:
960:
462:
176:. The game was first published under the name
1683:. Archived from the original on 29 June 2011.
1605:
922:, a set of 6 games played on hexavalent grids
134:. Hex was invented by mathematician and poet
2093:; downloadable material no longer available)
1822:
1820:
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2006:Hex Strategy: Making the Right Connections
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1395:
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1021:. N.Y., N.Y.: Simon and Schuster. pp.
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957:
681:
440:showed that Hex cannot be represented as a
1564:
1529:
1008:
1006:
1004:
1002:
1000:
626:First-player win, informal existence proof
489:and several universities) using a mix of:
34:
2112:with links to related mathematical papers
1854:Hex, inside and out : the full story
1829:Hex, inside and out : the full story
1817:
1582:
1507:Hex, inside and out : the full story
1429:
1379:
1358:
1273:
1128:Hex, inside and out : the full story
1077:Hex, inside and out : the full story
836:Projex is a variation of Hex played on a
487:Facebook Artificial Intelligence Research
1778:Pawlewicz, Jakub; Hayward, Ryan (2013).
1677:"Number of chess diagrams and positions"
1388:
566:
405:
356:
260:
216:that belongs to the general category of
196:marketed a version of it in 1952 called
1852:Hayward, Ryan B.; Toft, Bjarne (2019).
1827:Hayward, Ryan B.; Toft, Bjarne (2019).
1505:Hayward, Ryan B.; Toft, Bjarne (2019).
1175:
1126:Hayward, Ryan B.; Toft, Bjarne (2019).
1075:Hayward, Ryan B.; Toft, Bjarne (2019).
1012:
997:
989:Hayward, Ryan B.; Toft, Bjarne (2019).
145:It is traditionally played on an 11×11
2130:
2068:history, classification and complexity
1649:
1373:
1238:
1219:
743:Rectangular grids and paper and pencil
731:(also known as Gale and Bridg-It) and
714:Computed strategies for smaller boards
587:
1099:
361:A Parker Brothers edition of the game
1894:
1102:"The Lost Years of a Nobel Laureate"
1064:from the original on 9 October 2022.
773:
542:Relevant discussion may be found on
518:
422:
341:, who featured Hex in his July 1957
230:. Since the game can never end in a
1976:"Games and Puzzles 1973-08: Iss 16"
1796:from the original on 9 October 2022
991:Hex, Inside and Out: The Full Story
473:, published in the early 1980s for
309:, who introduced it in 1942 at the
13:
1999:
1978:. A H C Publications. August 1973.
1945:
1897:"How I invented games and why not"
1100:Nasar, Sylvia (13 November 1994).
700:quantified Boolean formula problem
584:evaluating of positions in chess.
352:
91:30 minutes – 2 hours (11×11 board)
14:
2194:
2034:
1674:
1532:The American Mathematical Monthly
1309:, Facebook Incubator, 28 May 2020
2043:free Net book by Matthew Seymour
1565:Even, S.; Tarjan, R. E. (1976).
1152:"nestorgames - fun to take away"
523:
1982:
1968:
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1780:"Scalable Parallel DFPN Search"
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463:Shannon's Hex-playing heuristic
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2143:Board games introduced in 1947
2138:Board games introduced in 1942
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1169:
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859:
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1:
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301:The game was invented by the
132:board made of hexagonal cells
2121:300dpi printable Hex board
296:
222:. It can be classified as a
203:
16:Abstract strategy board game
7:
1306:facebookincubator/Polygames
903:
847:
817:
722:
612:Brouwer fixed-point theorem
604:Symbols, Signals, and Noise
514:
436:In 1964, the mathematician
166:Brouwer fixed-point theorem
10:
2199:
1962:"ICGA – Computer Olympiad"
1473:10.1016/j.disc.2006.01.029
1190:10.1109/jrproc.1953.274273
880:
821:
806:
648:strategy-stealing argument
291:
208:Hex is a finite, 2-player
18:
2053:A Beginner's Guide to Hex
2016:(trade paperback, 363pgs)
1990:"Jeux & stratégie 06"
1787:Proc. Computers and Games
1057:. Parker Brothers. 1968.
831:
760:. According to the book
343:Mathematical Games column
245:. Hex can be played as a
103:
95:
87:
79:
71:
53:
45:
33:
21:Hex game (disambiguation)
2163:PSPACE-complete problems
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873:Hex is also part of the
682:Computational complexity
256:
180:in the Danish newspaper
2148:Abstract strategy games
2064:6 November 2020 at the
1681:John's Chess Playground
1492:21 January 2017 at the
864:As of 2016, there were
704:conjunctive normal form
453:In the 2000s, by using
226:, a particular type of
2173:Paper-and-pencil games
729:Shannon switching game
572:
362:
266:
243:Shannon switching game
214:abstract strategy game
62:Abstract strategy game
2168:Parker Brothers games
2041:Hex: A Strategy Guide
1895:Freeling, Christian.
1716:Victor Allis (1994).
1584:10.1145/321978.321989
838:real projective plane
824:Havannah (board game)
634:on any board of size
570:
544:Talk:Hex (board game)
406:Shannon's Hex machine
360:
264:
251:paper-and-pencil game
2183:Theorems in topology
2116:Printable Hex boards
1764:16 July 2011 at the
1735:2 April 2012 at the
1467:(19–20): 2515–2528.
1461:Discrete Mathematics
1013:Gardner, M. (1959).
896:Jeux & Stratégie
865:
802:
493:zero-learning as in
335:Princeton University
311:Niels Bohr Institute
2020:HEX: The Full Story
1885:Browne (2000) p.310
1876:Gardner (1959) p.78
1156:www.nestorgames.com
888:Games & Puzzles
630:In Hex without the
588:Mathematical theory
269:Hex is played on a
210:perfect information
161:of the game board.
30:
1650:Browne, C (2000).
1620:10.1007/bf00288964
1571:Journal of the ACM
1253:10.1007/BF00288964
1106:The New York Times
1051:Con-tac-tix manual
915:Mathematical games
573:
363:
267:
224:Maker-Breaker game
174:graph connectivity
123:) is a two player
28:
2028:978-0-367-14422-7
1946:Tapkan, M.Bedir.
1426:Browne, pp. 71–77
1417:Browne, pp. 29–30
1275:Kucherawy, Murray
875:Computer Olympiad
774:Rex (Reverse Hex)
656:Zermelo's theorem
602:in his 1961 book
561:
560:
467:alpha-beta search
423:Research timeline
125:abstract strategy
114:
113:
2190:
2158:Positional games
2153:Connection games
2078:dedicated to Hex
1994:
1993:
1992:. December 1980.
1986:
1980:
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1608:Acta Informatica
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1290:
1277:(January 1984).
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1241:Acta Informatica
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942:
910:Connection games
869:
763:A Beautiful Mind
556:
553:
547:
527:
519:
401:
400:
396:
393:
387:
386:
382:
379:
234:, Hex is also a
219:connection games
38:
31:
27:
2198:
2197:
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2192:
2191:
2189:
2188:
2187:
2128:
2127:
2066:Wayback Machine
2047:500 Hex Puzzles
2037:
2002:
2000:Further reading
1997:
1988:
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1766:Wayback Machine
1758:Solving 8x8 Hex
1756:
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1737:Wayback Machine
1728:
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1544:10.2307/2320146
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1494:Wayback Machine
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1184:(10): 1234–41.
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842:noncontractible
834:
826:
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811:
805:
791:
776:
754:
745:
725:
716:
696:PSPACE-complete
684:
628:
595:
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557:
551:
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541:
534:needs expansion
528:
517:
425:
408:
398:
394:
391:
389:
384:
380:
377:
375:
372:Parker Brothers
368:Parker Brothers
355:
353:Published games
337:. According to
327:
299:
294:
259:
236:determined game
228:positional game
206:
194:Parker Brothers
188:under the name
66:Connection game
64:
60:
41:
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17:
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2036:
2035:External links
2033:
2032:
2031:
2017:
2001:
1998:
1996:
1995:
1981:
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1938:
1913:
1887:
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1863:978-0367144258
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1598:
1577:(4): 710–719.
1557:
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868:over-the-board
861:
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807:Main article:
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442:binary matroid
424:
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412:Claude Shannon
407:
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339:Martin Gardner
326:
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305:mathematician
298:
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2059:Thesis on Hex
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1285:. p. 112
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569:
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552:November 2023
545:
539:
535:
532:This section
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438:Alfred Lehman
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212:game, and an
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119:(also called
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63:
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44:
37:
32:
26:
22:
2178:Solved games
2019:
2005:
1984:
1970:
1956:
1941:
1929:. Retrieved
1925:
1916:
1904:. Retrieved
1900:
1890:
1881:
1872:
1853:
1847:
1828:
1810:
1798:. Retrieved
1786:
1773:
1753:
1744:
1725:
1717:
1712:
1703:
1680:
1670:
1652:Hex Strategy
1651:
1645:
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1611:
1607:
1601:
1574:
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1535:
1531:
1525:
1506:
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1481:
1464:
1460:
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1445:
1422:
1413:
1408:Browne, p.28
1404:
1369:
1347:
1335:. Retrieved
1331:
1322:
1311:, retrieved
1305:
1299:
1287:. Retrieved
1282:
1269:
1244:
1240:
1234:
1225:
1221:
1215:
1206:
1181:
1177:
1171:
1159:. Retrieved
1155:
1146:
1127:
1121:
1109:. Retrieved
1105:
1095:
1076:
1070:
1050:
1016:
993:. CRC Press.
990:
940:
920:GIPF project
894:
886:
863:
855:
851:
835:
827:
812:
796:Blockbusters
794:
792:
788:Blockbusters
787:
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755:
746:
726:
717:
709:
685:
644:
639:
635:
629:
620:
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603:
596:
582:
578:
574:
562:
549:
538:adding to it
533:
479:
470:
459:
452:
449:
446:
435:
432:
429:
426:
410:About 1950,
409:
364:
328:
325:Nash's claim
318:
314:
300:
287:
280:
276:
268:
240:
217:
207:
197:
189:
181:
177:
163:
154:
150:
144:
120:
116:
115:
88:Playing time
49:1942–present
46:Years active
25:
2106:Game of Hex
2091:Hex archive
2087:Theory page
2030:(paperback)
1931:28 February
1279:"Hexmaster"
1161:3 September
860:Competition
752:Board sizes
688:Shimon Even
606:. In 1979,
593:Determinacy
477:computers.
475:Atari 8-bit
455:brute force
416:E. F. Moore
190:Con-tac-tix
2132:Categories
1906:19 October
1901:MindSports
1675:Tromp, J.
1449:Lasker, p.
1440:Browne, p.
1381:1801.05667
1360:2001.09832
1289:18 January
952:References
608:David Gale
247:board game
128:board game
80:Setup time
58:Board game
2110:MathWorld
1399:Browne p.
1111:23 August
768:John Nash
686:In 1976,
632:swap rule
510:AlphaGo).
495:AlphaZero
471:Hexmaster
347:Aage Bohr
331:John Nash
319:Politiken
307:Piet Hein
297:Invention
283:swap rule
204:Game type
182:Politiken
140:John Nash
136:Piet Hein
110:, tactics
2062:Archived
1922:"Projex"
1791:Archived
1762:Archived
1733:Archived
1689:cite web
1490:Archived
1198:51666906
1059:Archived
904:See also
848:Dark Hex
818:Havannah
809:Y (game)
723:Variants
515:Strategy
249:or as a
170:matroids
159:topology
108:Strategy
2072:HexWiki
1628:9125259
1593:8845949
1552:2320146
1261:9125259
881:Reviews
505:pooling
397:⁄
383:⁄
315:Polygon
292:History
271:rhombic
186:Denmark
178:Polygon
147:rhombus
72:Players
2026:
2012:
1860:
1835:
1800:21 May
1658:
1626:
1591:
1550:
1513:
1337:29 May
1313:29 May
1259:
1196:
1134:
1083:
1029:
832:Projex
780:misère
503:) and
303:Danish
192:, and
104:Skills
96:Chance
54:Genres
1794:(PDF)
1783:(PDF)
1624:S2CID
1589:S2CID
1548:JSTOR
1376:arXiv
1355:arXiv
1283:Antic
1257:S2CID
1222:JSIAM
1194:S2CID
1062:(PDF)
1055:(PDF)
1023:73–83
932:Notes
733:TwixT
501:U-Net
465:with
257:Rules
155:hexes
151:cells
2076:wiki
2074:, a
2024:ISBN
2010:ISBN
1933:2018
1908:2020
1858:ISBN
1833:ISBN
1802:2014
1695:link
1656:ISBN
1511:ISBN
1339:2020
1315:2020
1291:2019
1163:2020
1132:ISBN
1113:2017
1081:ISBN
1027:ISBN
778:The
690:and
483:here
414:and
388:-by-
232:draw
172:and
121:Nash
99:None
83:None
2108:at
2099:at
2097:Hex
1616:doi
1579:doi
1540:doi
1469:doi
1465:306
1249:doi
1186:doi
926:Tak
891:#16
702:in
333:at
198:Hex
153:or
117:Hex
29:Hex
2134::
1924:.
1899:.
1819:^
1789:.
1785:.
1760:,
1691:}}
1687:{{
1679:.
1622:.
1612:15
1610:.
1587:.
1575:23
1573:.
1569:.
1546:.
1536:86
1534:.
1463:.
1431:^
1390:^
1330:.
1281:.
1255:.
1245:15
1243:.
1226:12
1224:.
1192:.
1182:41
1180:.
1154:.
1104:.
1041:^
1025:.
999:^
959:^
899:#6
766:,
758:Go
739:.
737:Go
650:.
253:.
238:.
168:,
142:.
1964:.
1950:.
1935:.
1910:.
1866:.
1841:.
1804:.
1697:)
1664:.
1630:.
1618::
1595:.
1581::
1554:.
1542::
1519:.
1475:.
1471::
1384:.
1378::
1363:.
1357::
1341:.
1293:.
1263:.
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