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==Strategy==
==Strategy==
Although it is known that the first player (without the [[swap rule]]) has a theoretical winning strategy, it is not known what that strategy is, except for very small boards. However, there are a large number of useful tactical and strategic concepts available to Hex players.
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 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.<ref name="Browne p">Browne p.</ref>


===Virtual connections and templates===
[[Image:Hex situation bridge.svg|thumb|right|Diagram 1: bridge (A <--> C), a safely connected pattern]]


[[File:Some interior Hex templates.png|thumb|upright=1.2|Diagram 1: Some interior templates]]
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.<ref>Browne, p.28</ref> One of the simplest such patterns is the bridge (see diagram 1), which consists of two stones of the same color (A and C), and a pair of open spaces (B and D).<ref>Browne, pp. 29–30</ref> 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.<ref>Browne, pp. 71–77</ref> 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.<ref name="Browne p"/>
[[File:Some edge templates.png|thumb|upright=1.2|Diagram 2: Some edge templates]]


A set of stones of one color is said to be ''virtually connected'' if the stones' owner can guarantee to connect them, no matter what the opponent does. The simplest example of a virtual connection is a ''bridge'', shown in diagram 1. Although the bridge's two red stones are not adjacent, Red can guarantee to connect them: if Blue plays in one of the bridge's empty cells, then Red can play in the other. Note that the virtual connection requires not just the two red stones, but also the two empty cells of the bridge. The cells (empty or otherwise) that are part of the virtual connection are called the ''carrier'' of the virtual connection.
There are weaker types of connectivity than "safely connected" which exist between stones or between safely connected patterns which have multiple spaces between them.<ref name="Browne, p">Browne, p.</ref> The middle part of the game consists of creating a network of such weakly connected stones and patterns<ref name="Browne, p"/> which hopefully will allow the player, by filling in the weak links, to construct just one safely connected path between sides as the game progresses.<ref name="Browne, p"/>


A ''template'' is a pattern of stones and empty cells that is virtually connected and minimal (i.e., removing any stone or empty cell from the carrier would break the virtual connection). Templates can be characterized as ''interior templates'' (guaranteeing a connection between two or more stones) and ''edge templates'' (guaranteeing a connection between one or more stones and a board edge of the same color). Some examples of interior templates and edge templates are shown in diagrams 1 and 2, respectively.<ref>M. Seymour, ''Hex: A Strategy Guide'', online book, http://www.mseymour.ca/hex_book/</ref>
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.<ref name="Browne p"/> The skill is somewhat similar to the visualization of patterns, sequencing of moves, and evaluating of positions in chess.<ref>Lasker, p.</ref>


==Mathematical theory==
==Mathematical theory==

Revision as of 01:42, 19 July 2022

This article is about the abstract strategy game. For other uses, see Hex game (disambiguation).

Hex
11×11 Hex gameboard showing a winning configuration for Blue
Years active1942–present
GenresBoard game
Abstract strategy game
Connection game
Players2
Setup timeNone
Playing time30 minutes – 2 hours (11×11 board)
ChanceNone
SkillsStrategy, tactics

Hex is a two player abstract strategy board game in which players attempt to connect opposite sides of a rhombus-shaped board made of hexagonal cells. Hex was invented by mathematician and poet Piet Hein in 1942 and later rediscovered and popularized by John Nash.

It is traditionally played on an 11×11 rhombus board, although 13×13 and 19×19 boards are also popular. The board is composed of hexagons called cells or hexes. 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 topology of the game board.

Despite the simplicity of its rules, the game has deep strategy and sharp tactics. It also has profound mathematical underpinnings. The game was first published under the name Polygon in the Danish newspaper Politiken on December 26, 1942. It was later marketed as a board game in Denmark under the name Con-tac-tix, and Parker Brothers marketed a version of it in 1952 called Hex; they are no longer in production. Hex can also be played with paper and pencil on hexagonally ruled graph paper.

Game type

Hex is a finite, 2-player perfect information game, and an abstract strategy game that belongs to the general category of connection games.[1] It can be classified as a Maker-Breaker game,[1]: 122  a particular type of positional game. Since the game can never end in a draw,[1]: 99  Hex is also a determined game.

Hex is a special case of the "node" version of the Shannon switching game.[1]: 122 

Hex can be played as a board game or as a paper-and-pencil game.

Rules

Black vs white on a Hex board

Hex is played on a rhombic grid of hexagons, typically of size 11x11, although other sizes are also possible. Each player has an allocated color, conventionally Red and Blue or Black and White.[2] Each player is also assigned two opposite board edges. The hexagons on each of the four corners belong to both adjacent board edges.

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 board edges. The player who complete such a connection wins the game.

To compensate for the first player's advantage, the swap 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.

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.

History

Invention

The game was invented by the Danish mathematician Piet Hein, who introduced it in 1942 at the Niels Bohr Institute. Although Hein later renamed it to Con-tac-tix,[3][4] it became known in Denmark under the name Polygon due to an article by Hein in the 26 December 1942 edition of the Danish newspaper Politiken, the first published description of the game, in which he used that name.

Nash's claim

The game was rediscovered in 1948 or 1949 by the mathematician John Nash at Princeton University.[2][5] According to Martin Gardner, who featured Hex in his July 1957 Mathematical Games column, 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.[2] 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 Aage Bohr, 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.[6] 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."[1]: 134  In a later letter to Hein, Gardner added "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."[1]: 136 

Published games

A Parker Brothers edition of the game

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 11x11 empty board that could be played on with pencils or pens.[1] In 1952, Parker Brothers marketed a version of the game under the name "Hex", and the name stuck.[2] Parker Brothers also sold a version under the "Con-tac-tix" name in 1968.[3] Hex was also issued as one of the games in the 1974 3M Paper Games Series; the game contained a 5+1⁄2-by-8+1⁄2-inch (140 mm × 220 mm) 50-sheet pad of ruled Hex grids. Hex is currently published by Nestorgames in a 11x11 size and a 14x14 size.[7]

Shannon's Hex machine

About 1950,Claude Shannon and E. F. Moore constructed an analog Hex playing machine, which was essentially a resistance network with resistors for edges and light bulbs for vertices.[8] 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 attempting to solve the game and develop Hex-playing computer algorithms emulated Shannon's network to create strong computer players.[9]

Research timeline

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."[1]: 29 

It was also known to Hein that the first player has a theoretical winning strategy.[1]: 42 

In 1952 John Nash wrote up an existence proof that on symmetrical boards, the first player has a winning strategy.[1]: 97 

In 1964, the mathematician Alfred Lehman showed that Hex cannot be represented as a binary matroid, so a determinate winning strategy like that for the Shannon switching game on a regular rectangular grid was unavailable.

In 1981, the Stefan Reisch showed that Hex is PSPACE-complete.[10]

In 2002, the first explicit winning strategy (a reduction-type strategy) on a 7×7 board was described.

In the 2000s, by using brute force search computer algorithms, Hex boards up to size 9×9 (as of 2016) have been completely solved.

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 here). This program was based on Polygames[11] (an open-source project, initially developed by Facebook Artificial Intelligence Research and several universities[12]) using a mix of:[13]

  • zero-learning as in AlphaZero
  • boardsize invariance thanks to fully convolutional neural networks (as in U-Net) and pooling
  • and growing architectures (the program can learn on a small board, and then extrapolate on a big board, as opposed to justified popular claims[14] about earlier artificial intelligence methods such as the original AlphaGo).

Computer Hex

In the early 1980s Dolphin Microware published Hexmaster, an implementation for Atari 8-bit computers.[15] Various paradigms resulting from research into the game have been used to create digital computer Hex playing programs starting about 2000. The first implementations used evaluation functions that emulated Shannon and Moore's electrical circuit model embedded in an alpha-beta search framework with hand-crafted knowledge-based patterns. Starting about 2006, Monte Carlo tree search methods borrowed from successful computer implementations of Go were introduced and soon dominated the field. Later, hand crafted patterns were supplemented by machine learning methods for pattern discovery. These programs are now competitive against skilled human players. Elo based ratings have been assigned to the various programs and can be used to measure technical progress as well as assess playing strength against Elo-rated humans. Current research is often published in either the quarterly ICGA Journal or the annual Advances in Computer Games series (van den Herik et al. eds.).

Strategy

Although it is known that the first player (without the swap rule) has a theoretical winning strategy, it is not known what that strategy is, except for very small boards. However, there are a large number of useful tactical and strategic concepts available to Hex players.

Virtual connections and templates

Diagram 1: Some interior templates
Diagram 2: Some edge templates

A set of stones of one color is said to be virtually connected if the stones' owner can guarantee to connect them, no matter what the opponent does. The simplest example of a virtual connection is a bridge, shown in diagram 1. Although the bridge's two red stones are not adjacent, Red can guarantee to connect them: if Blue plays in one of the bridge's empty cells, then Red can play in the other. Note that the virtual connection requires not just the two red stones, but also the two empty cells of the bridge. The cells (empty or otherwise) that are part of the virtual connection are called the carrier of the virtual connection.

A template is a pattern of stones and empty cells that is virtually connected and minimal (i.e., removing any stone or empty cell from the carrier would break the virtual connection). Templates can be characterized as interior templates (guaranteeing a connection between two or more stones) and edge templates (guaranteeing a connection between one or more stones and a board edge of the same color). Some examples of interior templates and edge templates are shown in diagrams 1 and 2, respectively.[16]

Mathematical theory

Determinacy

John Nash was the first to prove (c. 1949)[17] that Hex cannot end in a draw, a non-trivial result colloquially called the "Hex theorem", which becomes known as equivalent to the Brouwer fixed-point theorem. Apparently, he didn't publish the proof. The first exposition of it appears in an in-house technical report in 1952,[18] in which he states that "connection and blocking the opponent are equivalent acts." The first rigorous proof was published by John R. Pierce in his 1961 book Symbols, Signals, and Noise.[19] In 1979, David Gale published a proof which also showed that it can be used to prove the two-dimensional Brouwer fixed-point theorem, and that the determinacy of higher-dimensional variants proves the fixed-point theorem in general.[20] A brief sketch of the no-draw ending requirement of Hex from that paper is presented below:

  1. Begin with a Hex board completely filled with hexagons marked with either X or O (indicating which player played on that hexagon).
  2. Starting at a hexagon vertex at the corner of the board where the X side and O sides meet, draw a path along the edges between hexagons with different X/O markings.
  3. Since every vertex of the path is surrounded by three hexagons, the path cannot self-intersect or loop, since the intersecting portion of the path would have to approach between two hexagons of the same marking. So, the path must terminate.
  4. The path cannot terminate in the middle of the board since every edge of the path ends in a node surrounded by three hexagons—two of which have to be differently marked by construction. The third hexagon must be differently marked from the two adjacent to the path, so the path can continue to one side or the other of the third hexagon.
  5. Similarly, if the sides of the board are considered to be a solid wall of X or O hexagons, depending on which player is trying to connect there, then the path cannot terminate on the sides.
  6. Thus the path can only terminate on another corner.
  7. The hexagons on either side of the line form an unbroken chain of X hexagons on one side and O hexagons on the other by construction.
  8. The path cannot terminate on the opposite corner because the X and O markings would be reversed at that corner, violating the construction rule of the path.
  9. Since the path connects adjacent corners, the side of the board between the two corners (say, an X side) is cut off from the rest of the board by an unbroken chain of the opposite markings (O in this case). That unbroken chain necessarily connects the other two sides adjacent to the corners.
  10. Thus, the completely filled Hex board must have a winner.

There is a reductio ad absurdum existence proof attributed to John Nash c. 1949 that the first player in Hex on a board of any size has a winning strategy. Such a proof gives no indication of a correct strategy for play. The proof is common to a number of games including Hex, and has come to be called the strategy-stealing argument. Here is a highly condensed informal statement of the proof:[2]

  1. It is impossible for the game to end in a draw (see above), therefore either the first or second player must win.
  2. As Hex is a perfect information game, there must be a winning strategy for either the first or second player.
  3. Let us assume that the second player has a winning strategy.
  4. The first player can now adopt the following defense. 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 way they play the winning strategy with one extra piece always on the board.
  5. This extra piece cannot interfere with the first player's imitation of the winning strategy, for an extra piece is always an asset[clarification needed] and never a handicap. Therefore, the first player can win.
  6. Because we have now contradicted our assumption that there is a winning strategy for the second player, we are forced to drop this assumption.
  7. Consequently, there must be a winning strategy for the first player.

Computational complexity of generalizations

In 1976, Shimon Even and Robert Tarjan proved that determining whether a position in a game of generalized Hex played on arbitrary graphs is a winning position is PSPACE-complete.[21] A strengthening of this result was proved by Reisch by reducing quantified Boolean formula in conjunctive normal form to Hex played on arbitrary planar graphs.[22] In computational complexity theory, it is widely conjectured that PSPACE-complete problems cannot be solved with efficient (polynomial time) algorithms. This result limits the efficiency of the best possible algorithms when considering arbitrary positions on boards of unbounded size, but it doesn't rule out the possibility of a simple winning strategy for the initial position (on boards of unbounded size), or a simple winning strategy for all positions on a board of a particular size.

Game tree of 11 by 11 Hex

In 11×11 Hex, there are approximately 2.4×1056 possible legal positions;[23] this compares to 4.6×1046 legal positions in chess.[24]

Computed strategies for smaller boards

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.[25] They extended the method to weakly solve the center pair of topologically congruent openings on 8×8 boards in 2002 and the center opening on 9×9 boards in 2003.[26] 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.[27] 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.[28] For every N≤10, a winning first move in N×N Hex is the most-central one, suggesting the conjecture that this is true for every N≥1.

Variants

Other connection games with similar objectives but different structures include Shannon switching game and TwixT. Both of these bear some degree of similarity to the ancient Asian game of Go.

Rectangular grids and paper and pencil

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 same way by using two different colored pencils.

Board sizes

Popular dimensions other than the standard 11x11 are 13×13 and 19×19 as a result of the game's relationship to the older game of Go. According to the book A Beautiful Mind, John Nash (one of the game's inventors) advocated 14×14 as the optimal size.

Rex (Reverse Hex)

The misère variant of Hex. 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 play can delay a loss until the entire board is full.[29] On boards with unequal dimensions, the player whose sides are further apart can win regardless of who plays first.[30] 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.[31][32] On boards with an even number, one of the first player's winning moves is always to place a stone in the acute corner.[29]

Blockbusters

Hex had an incarnation as the question board from the television game show Blockbusters. 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.

Y

Main article: Y

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.

Havannah

Main article: Havannah

Havannah is game based on Hex.[33] 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.

Projex

Projex is a variation of Hex played on a real projective plane, where the players have the goal of creating a noncontractible loop.[34] Like in Hex, there are no ties, and there is no position in which both players have a winning connection.

Competition

As of 2016, there were over-the-board 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. Hex is also part of the Computer Olympiad.

See also

References

  1. ^ a b c d e f g h i j Hayward; Toft (2019). Hex, Inside and Out: The Full Story. CRC Press.
  2. ^ a b c d e Gardner, M. (1959). The Scientific American Book of Mathematical Puzzles & Diversions. N.Y., N.Y.: Simon and Schuster. pp. 73–83. ISBN 0-226-28254-6.
  3. ^ a b Con-tac-tix manual (PDF). Parker Brothers. 1968.
  4. ^ Hayward, Ryan B.; Toft, Bjarne (2019). Hex, inside and out : the full story. Boca Raton, Florida: CRC Press. p. 156. ISBN 978-0367144258.
  5. ^ Nasar, Sylvia (13 November 1994). "The Lost Years of a Nobel Laureate". The New York Times. Retrieved 23 August 2017.
  6. ^ Hayward, Ryan B.; Toft, Bjarne (2019). Hex, inside and out : the full story. Boca Raton, Florida: CRC Press. pp. 127–138. ISBN 978-0367144258.
  7. ^ "nestorgames - fun to take away". www.nestorgames.com. Retrieved 3 September 2020.
  8. ^ Shannon, C. (1953). "Computers and Automata". Proceedings of the Institute of Radio Engineers. 41 (10): 1234–41. doi:10.1109/jrproc.1953.274273. S2CID 51666906.
  9. ^ Anshelevich, V. (2002). A Hierarchical Approach to Computer Hex.
  10. ^ Reisch, Stefan (1981). "Hex ist PSPACE-vollständig". Acta Informatica. 15: 167–191. doi:10.1007/BF00288964.
  11. ^ facebookincubator/Polygames, Facebook Incubator, 28 May 2020, retrieved 29 May 2020
  12. ^ "Open-sourcing Polygames, a new framework for training AI bots through self-play". ai.facebook.com. Retrieved 29 May 2020.
  13. ^ 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 (27 January 2020). "Polygames: Improved Zero Learning". arXiv:2001.09832 [cs.LG].
  14. ^ Marcus, Gary (17 January 2018). "Innateness, AlphaZero, and Artificial Intelligence". arXiv:1801.05667 [cs.AI].
  15. ^ Kucherawy, Murray (January 1984). "Hexmaster". Antic. p. 112. Retrieved 18 January 2019.
  16. ^ M. Seymour, Hex: A Strategy Guide, online book, http://www.mseymour.ca/hex_book/
  17. ^ Hayward, Ryan B.; Rijswijck, van, Jack (6 October 2006). "Hex and combinatorics". Discrete Mathematics. 306 (19–20): 2515–2528. doi:10.1016/j.disc.2006.01.029.
  18. ^ Nash, John (Feb. 1952). Rand Corp. technical report D-1164: Some Games and Machines for Playing Them. https://www.rand.org/content/dam/rand/pubs/documents/2015/D1164.pdf
  19. ^ Hayward, Ryan B.; Toft, Bjarne (2019). Hex, inside and out : the full story. Boca Raton, Florida: CRC Press. p. 99. ISBN 978-0367144258.
  20. ^ David Gale (1979). "The Game of Hex and Brouwer Fixed-Point Theorem". The American Mathematical Monthly. 86 (10). Mathematical Association of America: 818–827. doi:10.2307/2320146. JSTOR 2320146.
  21. ^ Even, S.; Tarjan, R. E. (1976). "A Combinatorial Problem Which is Complete in Polynomial Space". Journal of the ACM. 23 (4): 710–719. doi:10.1145/321978.321989. S2CID 8845949.
  22. ^ Stefan Reisch (1981). "Hex ist PSPACE-vollständig (Hex is PSPACE-complete)". Acta Informatica. 15 (2): 167–191. doi:10.1007/bf00288964. S2CID 9125259.
  23. ^ Browne, C (2000). Hex Strategy. Natick, MA: A.K. Peters, Ltd. pp. 5–6. ISBN 1-56881-117-9.
  24. ^ Tromp, J. "Number of chess diagrams and positions". John's Chess Playground. Archived from the original on 29 June 2011.{{cite web}}: CS1 maint: bot: original URL status unknown (link)
  25. ^ On a decomposition method for finding winning strategy in Hex game Archived 2 April 2012 at the Wayback Machine, Jing Yang, Simon Liao and Mirek Pawlak, 2002
  26. ^ Unpublished white papers, formerly @ www.ee.umanitoba.com/~jingyang/
  27. ^ Solving 8x8 Hex, P. Henderson, B. Arneson, and R. Hayward, Proc. IJCAI-09 505-510 (2009)
  28. ^ Pawlewicz, Jakub; Hayward, Ryan (2013). "Scalable Parallel DFPN Search" (PDF). Proc. Computers and Games. Retrieved 21 May 2014.
  29. ^ a b Hayward, Ryan B.; Toft, Bjarne (2019). Hex, inside and out : the full story. Boca Raton, Florida: CRC Press. p. 175. ISBN 978-0367144258.
  30. ^ Hayward, Ryan B.; Toft, Bjarne (2019). Hex, inside and out : the full story. Boca Raton, Florida: CRC Press. p. 154. ISBN 978-0367144258.
  31. ^ Gardner (1959) p.78
  32. ^ Browne (2000) p.310
  33. ^ Freeling, Christian. "How I invented games and why not". MindSports. Retrieved 19 October 2020.
  34. ^ "Projex". BoardGameGeek. Retrieved 28 February 2018.

Further reading

  • Hex Strategy: Making the Right Connections , Browne C.(2000), A.K. Peters Ltd. Natick, MA. ISBN 1-56881-117-9 (trade paperback, 363pgs)
  • HEX: The Full Story, Hayward R. with Toft B.(2019), CRC Press Boca Raton, FL. ISBN 978-0-367-14422-7 (paperback)


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