Talk:Hex (board game)

Move
OK, I moved this from Hex (game) to here as some anon IP did it by cut and paste. Rather than argue, I just deleted that and moved it properly. This does seem an acceptable place, ie we have Go (board game)... Evercat 21:16, 31 Aug 2003 (UTC)

Servers
is there a hex server somewhere on the net?

for example, on [deleted link], they have go, chess, checker, reversi, chinese checker.... and there's lots of go servers around. But never seen a hex server. P0lyglut 07:35, 2003 Nov 27 (UTC)

You can play Hex turn-based on [deleted link]. Reiner Martin 19:20, 27 Nov 2003 (UTC)

There are three realtime Hex servers that you might be able to find an opponent on. My direct links were deleted, so I replace them with names, which you could feed into a search engine.

A relatively popular server these days is Board Game Arena which is based in France. BGA offers grid sizes 6x6, 11x11, 14x14, and 15x15.

IGoogle Game Center may have stronger players. If you start a game table in all three servers, you may be more likely to snag an opponent.

Ludoteka is Spanish. They have a new client which offers odd sizes from 9x9 to 15x15. English menus are available. The board is oriented like a diamond instead of "tipped over" on the other servers. On Wednesdays a real time tournament takes place if there are enough participants.

The strongest field of players uses Little Golem which is turn based. That means your opponent is not necessarily logged on at the same time you are. Each game might take weeks, but you could play many games simultaneously.

--Twixter (talk) 08:45, 1 January 2017 (UTC) --Twixter (talk) 17:31, 2 April 2019 (UTC)

You can play Hex in real-time on [deleted link]. It supports different board sizes.

--artyomch 04:10, 21 Jun 2008 (UTC)


 * See [deleted link] —Kri (talk) 00:19, 22 January 2012 (UTC)
 * Please don't post gaming related links in wikipedia - Wikipedia is not a gaming site. Sbalfour (talk) 23:32, 24 January 2017 (UTC)

PSPACE vs. EXPTIME
I don't understand the connection between PSPACE and EXPTIME with the level of difficulty. Perhaps this should be made clearer in the article. --Johnleemk 14:05, 3 May 2004 (UTC)

PSPACE and EXPTIME are terms in computational complexity that measure the difficulty of a problem (how long it takes to solve a generic instance of the problem in terms of the size of the input). I added a few sentences to clarify. Basically, it means that we know that generalized chess is at least as hard as generalized hex, but most experts think chess is probably harder. --Ptrillian 22:24, 1 January 2007 (UTC)

Tie games
Why can the game never end in a tie? It seems plausible, but how does one prove it? pstudier 06:52, 2004 May 27 (UTC)

OK, suppose the game is over, and neither player won. Then the whole board is filled with your stones and your opponents stones, or the game wouldn't be over. Now consider one of your home edges, and all of your stones connected to that edge. That mass of stones doesn't reach your other home edge, or else you would have won. So consider the frontier of that mass of stones. All of the frontier must be either your opponent's stones, or your opponent's edges, because the whole board is filled. That frontier must connect your opponent's two home edges (a topological fact unless the board has holes). But that means he must have won: a contradiction. Therefore the original assumption that the game was a tie is false. This isn't a formal proof, but it may be enough for informal conviction. --Fritzlein 19:44, 28 May 2004 (UTC)


 * The problem with this argument is that it would imply that a board made of squares would also not end in a draw. But, if you imagine a board filled in like a checkerboard, you'll see that the square version can end in a draw. I've added a sketch of David Gale's proof that Hex always has a winner, which requires the geometry of hexagons to work. --2606:6000:6107:4500:4ED4:BF52:CF56:5082 (talk) 11:01, 14 November 2018 (UTC)

Another way to look at it (more informal than the version Fritzlein gave) is to try to build a tie in an actual board, or think what a tie must be. You'll see quite easily that it is impossible. RBerenguel 09:40, 9 November 2007 (UTC)

The article is underselling Nash's contribution. Nash showed that not only must the game have a winner (which is relatively easy to verify on these space filling games), but that there is a winning strategy for the first player. i.e. if player 1 plays a perfect game he will win. —Preceding unsigned comment added by 194.159.45.130 (talk) 15:09, 12 June 2009 (UTC)

Independently invented, not rediscovered
"REDISCOVERED and popularized by John Nash" -- I believe this is a rewrite of history. As I've heard, Hex was independently invented by both Nash and Hein. Neither had heard of each other and were unaware of each others' game. If I can find precise references to this, I'll be sure to post them here. 97.113.65.10 (talk) 20:00, 14 October 2023 (UTC)


 * There is more about it in this section: https://en.wikipedia.org/wiki/Hex_(board_game)#Nash's_claim
 * I had also heard that it was an independent invention, but with the publication of HEX: The Full Story, new information came to light, or so it seems. 2003:DF:3F07:4400:4AF6:2CD8:7FBC:CB8F (talk) 04:51, 9 November 2023 (UTC)

Game board graphic
Is the giant Go board really necessary on this page? This should be describing the game Hex itself, not the way a particular image of a board has been created. --ambience 04:16, 9 Jan 2005 (UTC)

First of all, that was a Hex board, not a Go board. Each internal vertex had six lines radiating from it, not four. The board was a diamond shape, not a rectangle. And did you see the border colors?

I apologize for including my description of how it was made. I apologize if the image was too large to suit you. I could make a smaller image if there is interest. Does anyone besides me think that such an image does indeed describe the game of Hex? Twixter 16:50, 27 Feb 2005

I don't have a problem with the image. It was kind of large when it was horizontal, so I rotated it a while back, but it seems appropriate. If it's still too large you could try specifying a certain pixel width in the article tag. Might make it less jaggy too. DreamGuy 04:44, Feb 28, 2005 (UTC)

DreamGuy, you and I are talking about two different images. I created a POV-ray perspective image of a Hex board. It has been removed. You can see this image in the previous revisions of the Hex (board game) page. I now have a smaller version of the image:



I would be glad to include it on the main page for Hex, without the extraneous discussion I foolishly added the first time. But first, I would like to know if anyone would be interested in it besides me.

And regarding what you did to the other image, I'm sure you meant well, but when two players sit down for a face to face game, they don't orient the board like that. An image serves other purposes than simply fitting on the page well. Some players might like to rotate the board 30 degrees from horizontal, since that is how several servers display it, but not 90 degrees for heaven's sake! That vertical image is disturbing to the eyes of any Hex player. I urge you to rotate it back again.

--Twixter 17:57, 4 Mar 2005 (UTC)


 * Hrm, I somehow missed the other one showing up and being removed. Doesn't yours have too many spaces? And the way it looks it could be a square board in perspective. Regarding the orientation of the graphic currently there, it's a top view, not a view from where someone playing would be sitting... players don't hover 5 feet above the board when they play either way, right? DreamGuy 18:59, Mar 4, 2005 (UTC)

How many spaces is too many? It's a 19x19 grid. That's not too many for the 282 players who have enjoyed playing 19x19 on Little Golem. Should I repeat myself? There are six lines radiating from each interior vertex. Perhaps "the way it looks" depends on how closely you look at it. Either you like it or you don't, I guess. No reason is necessary.

Yes, the current graphic is a top view. So are the graphic displays on Little Golem, Kurnik, and all the other Hex servers, both real-time and turn-based. All of them show the board oriented so the long diagonal is either horizontal or 30 degrees off horizontal. That's how the vast majority of Hex games are played these days: on the Internet. These vertically oriented images are, well, disturbing. In my opinion they do not describe the game of Hex, as it is played by so many people, as well as it would if they were rotated back. --Twixter 14:29, 8 Mar 2005 (UTC)

I think we lack a user-box capable graphic, in my user page I just put a hexagon, but it lacks some visual appeal. Does anybody mind adding some kind of Piet Hein problem (the one 3x3 would be dear) to this page? This way we could use it to create an user box. RBerenguel 16:24, 9 November 2007 (UTC)

Re-word the rules section? And replace the image there?
The Rules as currently stated here are not wrong, but IMO they could be stated more clearly. For example, emphasis is made on completing your path before your opponent completes theirs. This may mislead readers into thinking this is a pattern-forming game like Gomoku. Such games have a "racing" aspect to them. Hex is a blocking game, not a racing game. The only way to form your path is to block your opponent's path. Once you have created a pattern which blocks the opponent, there will be nothing they can do to stop you from completing your path.

Another reason I want to re-word the rules has to do with comparisons to Go. Many Go adherents claim that their game has the highest "ratio" of emergent game complexity to rules complexity. While I personally agree that 19x19 Hex is probably not as deep as Go, the rules to Hex are significantly simpler. One benchmark I like to use when arguing my case is how the rules to these two games are stated in Wikipedia. The Go rules currently consist of eight sections (including the beginning of the rules) and eight diagrams. Here is my proposal for the rules of Hex. It restates information already provided at the beginning of the article (which I do not propose to change,) as well as a description of the pie rule which is available in the provided link, but I want to put the complete rules from scratch in a single contiguous block, to show that they are still much simpler than the Go rules.

The board is a hexagonal grid of cells, traditionally in the overall shape of a rhombus. The cells could be represented as a grid of hexagons, where stones are placed in the hexagons, or as triangles, where stones are placed on the intersections. The rows of cells around the edges are called border rows. Each of the four corner cells are part of both adjacent borders. Two players have their own color collection of stones, for example black versus white. Before the game starts, the board is empty, and each pair of opposite border rows is designated as belonging to a specific player. With one exception, each move consists of placing a single stone of your color in any vacant cell. The object is to make a continuous path of your color stones which connects your border rows.

There is one more rule, called the pie rule, or the swap rule, or one-move equalization. Suppose black moves first. At the point in the game when black has placed the first stone on the board, the second player has the option to swap. There are two possible ways to implement this swap move. The second player could exchange their container of stones with the opponent. This is called swapping colors. The player who placed the first stone on the board as black becomes white, and makes the next move. Alternatively, the white player could place a white stone in a cell which is in a mirror image location to the black stone's cell, reflected across the nearest board diagonal, either the short diagonal or the long diagonal (which does not consist of contiguous cells.) If black's first stone is on a board diagonal, the white stone could replace the black stone there. Then the black stone is returned to black's container. This is called swapping the first stone. In terms of game play, this is effectively the same as swapping colors. Swap may be done only once per game. If the second player chooses to place a stone instead of swapping after the first stone is placed, then swap may not occur at all in that game. This rule makes Hex much more fair and deep. It is called the pie rule because it is like when two people want to share the last of the pie. One person cuts the pie into two slices, and the other chooses which slice to eat. Of course either player is allowed to resign before a winning path is completed. A drawn position can never occur, but several online servers allow agreed draws.

I am certain the creator of the previous image would not object, because that person is me. I welcome all feedback. Nit picking is encouraged. I want to get this right. My plan is to wait a couple months before I edit the main page. I don't want to enter into an editing battle. --Twixter (talk) 20:39, 4 April 2019 (UTC)

Regarding the diagram: while I like (and actually prefer) the "go style" boards with lines, the convention for Hex is the board with hexagonal tiles. It's the form used by Hein when he first introduced the game, as well as by Nash and Gardner in their articles, and more recently Browne and Hayward in their books. It's also the convention used in academic articles, most websites where Hex is played, and many physical sets (there was a "holes and pegs" version of Hex published, but I'm not aware of any published go style Hex boards). Given that this is overwhelmingly considered the convention for Hex boards, I think the Wikipedia article should stick with that convention as well. 134.134.139.72 (talk) 18:51, 5 April 2019 (UTC)

Brouwer fixed-point theorem
In sect. Theory and Proofs one reads

An important consequence of the determinacy of hex is the Brouwer fixed-point theorem which was shown by David Gale.

I'm not sure I fully understand what is meant, but it is clear that the Brouwer fixed-point theorem is not a consequence of the hex game (and this theorem was not "shown by Gale" but by Brouwer) ; maybe the other way round ? &mdash; MFH:Talk 12:20, 24 October 2006 (UTC)


 * I had the same question but it checks out; I read through a summary of the results in the cited article (American Mathematical Monthly) on MathSciNet. Gale's proof was certainly not the first nor the most important, but he does show equivalence of the two theorems.  The fixed-point theorem in n dimensions requires a different but related n-player game.  I've tried to make this clearer in the article.  Tracy Hall 17:55, 22 March 2007 (UTC)

Examples and more details for Template Section
I'm a new contributer to wikipedia, so I'm still trying to learn the nuances of what other contributers believe the purpose of wikipedia is. I think it would be a good idea to add examples of some templates (at least 5 or 6 of the smaller ones) along with the analysis to explain why a template is connected. I like one explanation I read (perhaps in Browne's book, I don't recall) of 2-connectedness. The reason a template (or any winning position) works can usually be explained well in terms of being 2-connected. I.e. "if he moves there, then I move here to stay 2-connected." Do others agree that this would be a useful addition? If so, does anyone have ideas about a good way to create the images of the templates. I could probably write up a decent explanation, but it wouldn't do much good without pictures. What do people think? --Ptrillian 21:55, 1 January 2007 (UTC)

Certainly 2-connectedness is a very basic concept which is essential to nearly all Hex strategy. I would prefer not to submit any more images to Wikipedia, but it shouldn't be difficult for someone else to use a Hex tool such as Jhex or Ohex, an image capture tool, and an image editor to show whatever patterns or templates you wish to. Please be sure to verify that your image is your own creation and that you release it to the public domain, when you upload it.--Twixter 02:50, 12 February 2007 (UTC)

Future FA?
I think this article has much room for adding new things. I can imagine tons of different images and mathematical information. I feel that it has the potential to be an FA. Compare with chess, a featured article. Below are the subtopics from the chess article and how a corresponding section could be made for hex.


 * Rules - got that.
 * History - easy to expand. The current article does not include information about anything past the year 1952, except for the mention of Cameron Browne's Hex Strategy.
 * predecessors - Hmm...
 * origins - Hmm...
 * birth of a sport
 * post-war
 * Place in Culture - well... Place in Mathematical Culture?
 * Notation for Recording Moves - Easy to add.
 * Strategy and Tactics - enough info about this out there, I believe
 * fundamentals of strategy - doable.
 * fundamentals of tactics - I think templates, ladders, and such belong here.
 * opening - easy to add, already have some external links
 * middle game
 * endgame - I think the closest thing to an endgame in hex is where one player realizes he/she has a winning connection. But I still think it's possible to write a blurb about that.
 * chess composition - there are a few problems on HexWiki that are usable
 * competitive play
 * organization of competitions - mention online playing sites like kurnik, boardspace, littlegolem; mention the ICGA
 * Titles and rankings - well... according to HexWiki, "champion" on littlegolem is currently the most prestigious title
 * Mathematics and computers - mention programs, algorithms; solutions for the game without the swap-option...
 * psychology - nothing.
 * Variants - got that

Anyone agree?

Leon math 18:25, 9 April 2007 (UTC)
 * The problem is that there's only a single book ever written on Hex, and it's not by a recognized champion or game authority. Nor is there any magazine where Hex is regularly or even periodically featured. There's no commercial publication of the game (the Parker Brothers and Danish editions are long out of date and unavailable). There's little sponsored competition and no references for exemplary games to illustrate play of the game.  The history of the game is largely confined to its invention.  It's an exceedingly rare game in play; most of the interest is research related topology, graph theory and combinatorics.  Those are arcane topics to go into in a board game article.


 * The article as it stands today, after much reoganization, additions, and citations, is barely 'B' class (though it was undeservedly classified 'B' before); the problems remain coverage, citations, and a large chunk of probably original research (though might be able to cite Browne's Hex book). We're a long way from FA, need to focus on getting the basics of a concise scholarly article first. Sbalfour (talk) 18:04, 24 January 2017 (UTC)

board representations
That board is just like a go board except with diagonal lines drawn in one of two directions. Does this mean hex can be played with a go board?

68.7.25.121 (talk) 06:38, 28 July 2008 (UTC)srn347

If you start with a go board and then add one set of diagonal lines, you end up with some adjacent directions at 90 degrees and some at 45 degrees. The diagram here, however, shows all adjacent directions at 60 degrees.

You can add lines as you describe and play Hex on a modified Go board if you wish, certainly. For that matter, there are probably players who can play blindfolded, without the use of a set. I certainly cannot, and I greatly prefer a board which either consists of hexagons, where stones are played inside the hexagonal cells, or triangles, where stones are played on the intersections. The vast majority of sets, both physical and virtual, use hexagons.

It is possible to play a game similar to Hex on a standard Go board, but the rules to this different game must deal with the potential confusion regarding what cells are adjacent to what other cells. --Twixter (talk) 17:47, 2 April 2019 (UTC)

Cameron Browne's book
The article said: [Cameron Browne]] wrote a book entitled Hex Strategy: Making the Right Connections'', which covers Hex strategy at a greater level of detail than any preceding work. However, some Hex players feel that this book contains many factual errors and advocates questionable strategies .''

However, if we check the sources, they say e.g. Players have questioned some of the advice given. This is completely different to his book contains many factual errors and advocates questionable strategies. The other source says basically that there are things Browne does not mention. That is even less support for the wording.

I have taken the whole paragraph out and replaced it with a pointer to the existence of Hex strategy books. Going beyond that is hardly relevant. -- Zz (talk) 16:39, 15 November 2008 (UTC)

I think there is only one book specifically dealing with Hex strategy, and although it may not be perfect, I'd like to see its name or some way for user to find it via wikipedia (of course one could google "hex book"). --Halladba (talk) 15:38, 18 November 2008 (UTC)

Ok. A game matures when books about it hit the market. I am not against mentioning it anymore. -- Zz (talk) 19:14, 20 November 2008 (UTC)

Nash and Connection games sub-genre
Did Hein and Nash invent not only Hex but the entire sub-genre of connection games which require side-to-side connections to win (Hex, Twixt, Gonnect, etc)? —Preceding unsigned comment added by 207.102.64.208 (talk) 07:17, 15 February 2009 (UTC) This mechanism in itself, one gathers, is totally original and unprecedented, but this has never been stated clearly in anything I have read. —Preceding unsigned comment added by 207.102.64.208 (talk) 07:23, 15 February 2009 (UTC)

14x14?
Did Nash ever give any reason for why he considered 14x14 the optimal size? —Preceding unsigned comment added by 70.182.69.205 (talk) 03:33, 8 May 2010 (UTC)

Go strategy
The claim that Nash thought the winner of Go was determined by chance and not strategy is in serious need of a credible reference; Go is quite obviously a game that involves an enormous amount of strategy - a fact well known among computer programmers and laypersons alike. I will be very surprised if the man who invented game theory held such an obviously fallacious opinion about Go. Failing the addition of a credible citation, this claim (pardon the pun) needs to Go! Spiral5800 (talk) 10:22, 18 February 2011 (UTC)
 * Indeed it requires a citation! But note (1) that game theory has little to do wth games like go and hex (which of course doesn't mean that Nash wouldn't be aware that go is a strategy game with perfect information), and (2) go is - unlike hex - so complex, so far outside what any supercomputer could analyse, that among roughly equal players (or players matched with a suitable number of handicap stones), the outcome at the end of a long game has for all practical purposes a substantial element of randomness. I'm not saying the claim in the article is right, or that the claimed claim by Nash is right - but that it might be, in a sense.--Nø (talk) 11:57, 18 February 2011 (UTC)
 * Certainly seems unlikely to have been worded in these terms. I would see little problem with you removing the comment and cleaning up the history section. Especially since "invented indepedently" is not the same as inventing the game.Tetron76 (talk) 18:33, 1 March 2011 (UTC)

First-player strategy
Who has the winning strategy when the pie rule is used? (My guess would be the second player, since he can "become" the first player if he wants, and he has a winning strategy wherever the first stone is placed for him.) If anyone knows could they add it to the article? — Preceding unsigned comment added by 92.27.55.215 (talk) 21:10, 23 May 2012 (UTC)
 * You are correct: the second player can win under the 'pie' rule. The first player can make a move in either of two sets: moves which maintain his winning advantage, and those that do not.  If both players play perfectly, and the first player makes a winning move, the second player will swap the board; if the first player makes a losing move, the second player will let him keep it. However, no players are good enough to play perfectly.  The real purpose of the rule is to cause the first player to make a weak but plausable move, to equalize the winning changes of (imperfect) players.  Sbalfour (talk) 19:45, 20 January 2017 (UTC)

first proof of PSPACE completeness
The first to prove that HEX is PSAPCE complete were Shimon Even and Robert Tarjan. The proof was published in S. Even and R. E. Tarjan. 1976. A Combinatorial Problem Which Is Complete in Polynomial Space. J. ACM 23, 4 (October 1976), 710-719. DOI=10.1145/321978.321989 http://doi.acm.org/10.1145/321978.321989 --79.176.112.80 (talk) 21:20, 15 October 2013 (UTC)

2620:10D:C093:200:0:0:1:A530 (talk) 14:57, 18 February 2020 (UTC)

Viewing more of that paper than the first page seems to require an account or payment, so it's remotely conceivable that further in it shows PSPACE-hardness for Hex, but all indications from the abstract and the first page are that it only shows hardness of vertex Shannon games. JumpDiscont (talk) 11:26, 12 April 2021 (UTC)

I believe the full paper is here: http://www.cs.technion.ac.il/users/wwwb/cgi-bin/tr-get.cgi/1975/CS/CS0060.pdf Hex is equivalent to the vertex Shannon switching game (on a particular grid), which appears to be the version this paper discusses. 24.52.230.205 (talk) 14:49, 16 April 2021 (UTC)

Computers vs humans

 * For a while 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 (game available here: https://littlegolem.net/jsp/game/game.jsp?gid=2129789, arxiv paper explaining the method: https://arxiv.org/abs/2001.09832). The program has the property that it can learn on a small board, and then extrapolate on a big board, contrarily to popular claims about earlier artificial intelligence methods (see the widely cited https://arxiv.org/pdf/1801.05667.pdf). Disclaimer: I am one of the authors of that program (O. Teytaud), hence I did not edit, just posted this in the present discussion page.

HexWiki a wiki dedicated to Hex
This external link is down:

We currently have some problems with the database. Please come back in a few daysweeks. The provider has been unable to fix his database for the last three months.

Does it take long? How do i got the documentation back? — Preceding unsigned comment added by Bart veurink (talk • contribs) 14:37, 4 January 2014 (UTC)
 * Your promotional link to an external wiki unrelated to Wikipedia has been deleted (apparently), and properly so. Wikipedia is not responsible for that site, and the talk page of this article is not a forum for general discussion of hex or playing hex.  It is solely for discussion of content and editing of the article. Sbalfour (talk) 21:53, 24 January 2017 (UTC)

John Nash apparently did not independently invent Hex
https://boardgamegeek.com/blogpost/59630?commentid=6742048#comment6742048

If true, a lot of references to Hex across the Internet, not just this page, would need some serious editing. But the above reference is not solid enough for that IMO. If you, dear reader, happen to know more about this, or can afford to access the ICGA journal, 38:2, pp.126-7, any verifiable detail would be appreciated. --Twixter (talk) 08:11, 1 January 2017 (UTC)
 * I'd play this like newspaper sourcing: stories, especially sensational ones, must have at least a second corroborating source. Sbalfour (talk) 19:54, 20 January 2017 (UTC)

Here are two relevant Martin Gardner Sci Am sources:

Martin Gardner's article first popularizing Hex: Gardner, Martin 1957. "Mathematical Games" Scientific American 197, no.1, July 1957, pp. 145-146, 148, 150. Accessed September 25, 2017.  

In this first article, Piet Hein is credited as the inventor, having presented it at the Neils Bohr institute in 1942. Neils Bohr's son, Aage, then introduced the game at the Institute for Advanced Study in Princeton in 1949. The game was said to be called "John" by Princeton students because it was played on hexagonal tiles on bathroom floors. However, later the same year, Gardner writes of John Nash's independent invention of the game in 1948.

Later issue mention of John Nash's independent invention: Gardner, Martin 1957. "Mathematical Games" Scientific American 197, no.4, October 1957, pp. 138 Accessed September 25, 2017. 

So there are two possible lines of interpretation as I see it: (1) Piet Hein presented it at the Neils Bohr Institute (while Neils Bohr still lived), then through the Institute or from his father Neils, Aage learned the game and communicated it to Princeton where it then was learned of and analyzed by John Nash. (2) John Nash independently devised the game, given his interest in Go and the hexagonal tiles on Princeton bathroom floors. The date of Nash's reported invention seems so close to Aage's propagation of the game to Princeton that it would seem that Piet Hein is likely the sole inventor. I would be interested in which sources support Nash's independent invention given the Piet Hein -> Neils Bohr Institute -> Neils Bohr -> Aage Bohr -> Princeton Institute for Advanced Study connection that would also explain Hex's popularity in Princeton in the late 1940s. The Wikipedia article on Aage Bohr notes that "In early 1948, Bohr became a member of the Institute for Advanced Study in Princeton, New Jersey.", i.e. Aage arrived on the scene at Princeton early in the year Nash is said to have independently invented Hex.

Unless we can find solid evidence of Nash's independent invention, all other evidence points to Piet Hein as the sole independent inventor.

Todd (talk) 23:05, 11 November 2017 (UTC)

It should also be noted that David Gale, who heard of Nash's proof, didn't credit Nash as an "inventor", but as one who "rediscovered" the game. This to me would indicate that Gale didn't think Nash invented it, but rather discovered and promoted mathematically significant insights with regard to it. See Sylvia Nasar's remarks quoted here: Todd (talk) 00:54, 12 November 2017 (UTC)

There's a new book for sale, HEX The Full Story by Ryan Hayward with Bjarne Toft. This should definitively resolve the issue of Nash's contribution. --Twixter (talk) 20:05, 2 April 2019 (UTC)

I added information on Nash's claim of independent reinvention with information in Hayward's book. I followed with Gardner's conclusion: there is reason to believe that Nash may have been unknowingly influenced by Hein's invention, but that Nash's claim, with no direct refutation, should be taken seriously as he was a respected mathematician. Pretentieux (talk) 22:41, 18 October 2020 (UTC)

Shannon switching game/Gale/Bridg-It
These games aren't really derivatives or variants of Hex; the nature of their connectivity is different (hex is 6-connected, Shannon's game is 4-connected generally) and this gives rise to profound differences in proofs and strategy: the Shannon game is representable as a binary matroid, and is solved. Hex is not a binary matroid; the strategy is very much more complex. I think I'm going to pull Gale/Bridg-It into a separate article, or merge these subsections into the article on the Shannon switching game, which they ARE closely related to, as in synonymous.Sbalfour (talk) 15:33, 15 January 2017 (UTC)


 * Agreed they're (it?) are not variants of Hex. Or derivatives — possibly. (But David Gale *was* around Princeton and knew John Nash when Hex was invented.) Nevertheless, they are very closely related kinds of topological / combinatorial games. For this reason, Bridg-It et al. might best be included toward the end of this article under a section called Related games. If the material got to be too large, it would merit its own article/s.2600:1700:E1C0:F340:597F:5057:E89D:168E (talk) 01:05, 30 April 2019 (UTC)

Strategy section
The Strategy section is bare bones, more like an example section, like showing a castled king position or checkmate position in chess. It borders on WP:ORIGINAL RESEARCH. We might be able to cite Cameron Browne's book. But if we go much further, we'll have a WP:GAMEGUIDE. Take a look at historical versions of the TwixT article for what NOT to do. I'm at a loss for how to make this section useful as well as terse and scholarly. In chess, there's openings, endings, combinations and position play. Maybe that'd be a good outline here.Sbalfour (talk) 21:54, 17 January 2017 (UTC)

I've added a bird's eye view description of the strategy, by breaking the objective of the game, to complete a chain between sides, into three steps or phases of the game. I adopted an unfortunate term 2-connectedness for an inherent property characterizing the phases, because that term is already used in the text (borrowed from its usage in the literature) to mean a path between points which contains two open cells. My terminology is original; a similar term by Anshelevich is "virtual connection", i.e. virtual connectedness, though he uses the term only in a local sense as a reduction element in heuristic search. Maybe dual-connected could be substituted to resolve the ambiguity.

Even if the terminology were changed, my argument itself contains an ambiguity born of terseness and simplicity of statement. The connectivity of the open board at the first move is not of the same kind as 2-connectedness: 2-connectedness implies that the player for whom it exists could "pass" if that were permitted by the rules (in practice, he could make a random move) and still win. That is not true on the open board: the first player must make a move from a small set (at least 11 cells on the short diagonal and possibly 25 or more clumped around it) in order to preserve a win. Nor is it exactly 1-connected, where the putative path between points (or edges of he board) requires filling in one cell to join two chains, i.e. the set of cells to be played contains a single cell. 2-connectedness is analogous to the connectivity of the Shannon switching game, wherein the board can be inscribed with two edge-disjoint spanning trees. The connectivity of the open hex board cannot be inscribed with a pair of such trees - the initial part of the game consists of reducing that connectivity to the Shannon-type connectivity. Yet, even that may not be enough for recognition of the 2-connected property: the corresponding edges of the dual trees may not enjoy a simple adjacency propiquinty, but can be remote from each other. Elaborating the stated argument loses something in brevity and elegance for precision. It is in a more general sense true that the hex board contains a property of complex serial connectivity analogous to the binary static connectivity of the Shannon game, which is to be preserved (or broken in the view of he second player). It is also true that that complex connectivity may not be resolved into 2-connectedness during the course of the game, though it almost universally occurs well before a recognized end to the game. It is precisely the nature of this connectivity which is the unsolved aspect of hex; it may be that solution of hex depends on defining how that connectivity can be resolved into 2-connectedness given the alternating turns of play. Sbalfour (talk) 16:18, 24 January 2017 (UTC)

I've replaced the strategy section, in line with similar (brief) sections in other articles. The strategy section is now colloquial rather than formalistic with definitions, etc. The phrasing contains a minimum of game jargon. It is more like a description of game play of interest to a general audience: is the game thoughtful (chess) or casual (yahtzee); how are the board and game pieces used; are there characteristic phases or maneuvers during play; what skills are required? It contains a minimum of game-specific details. Finally, the strategy section is very brief - it doesn't swamp the article presentation space, or distract attention from the rest of the article with glitzy diagrams, tables or illustrative positions from the game. This type of description could comfortably be fitted into a Description or Gameplay section, obviating the need for a separate Strategy section altogether. It should be emphasized that no strategy section is required for GA and except in rare cases (chess is one) for FA. Its presence is an adjunct to a concise but complete scholarly article, not a constituent part of it. If an article would suffer for its deletion, the article and strategy section need to be restructured so all essential information is integrated elsewhere. Sbalfour (talk) 17:29, 28 January 2017 (UTC)

Theory and proofs: strategy stealing agument
The argument has been badly botched. Consider for example the simple statement that if it's been proven that the game can't end in a draw, then one player or the other has a winning strategy. Suppose that the markers are placed on the board alternately by lot. By the time the 121st marker has been placed, one player or the other MUST have won. But neither player has a winning strategy! That is, neither player can FORCE a win. Sheeesh. The argument either needs to be deleted entirely (just say it's been proven that the first player can win), or write that argument in correct prepositional logic.Sbalfour (talk) 23:40, 17 January 2017 (UTC)

I've deleted it; it takes precise exposition to state, and I'm just not into it right now. Maybe a mathematician/editor will be able to reinstate this.Sbalfour (talk) 18:55, 19 January 2017 (UTC)

I've restored the argument from Gardner's rendition of it in Scientific American. It's still very informal, but considering the informal presentation of the article itself, it'll do for now. Sbalfour (talk) 22:00, 24 January 2017 (UTC)

John Nash said?
If John Nash said or proved something, it must be sourced - a citation to a peer-reviewed professional journal. Otherwise, the statement cannot remain in the article. I'm removing all such unsourced or improperly sourced statements.Sbalfour (talk) 06:05, 18 January 2017 (UTC)

Variants section
Trying to ref the Variants section, it appears that someone perused the IG Games website and copied all the Hex-like games to here. Easy effort, no scholarship. The list isn't unduely long; the issue is the notability of the items on it. Wikipedia isn't an almanac or gamesite. I propose the following criteria for including/excluding items from the list:

Inclusion criteria (at least one of):


 * 1. there is a published book devoted to the game, or at least a whole chapter in a book devoted to the game
 * 2. there is at least one paper in a peer-reviewed professional journal devoted exclusively to the game
 * 3. there is a commercially published game product
 * 4. there's a registered patent, trademark or copyright for the game even if it isn't published
 * 5. there's regulated national or international level competition in the game
 * 6. there's a wikipedia article for it (somewhat dubious - there's a lot of wiki articles on non-notable things)
 * 7. it's listed on the Game Geek website (somewhat dubious)

Exclusion criteria (any single one excludes)


 * 1. games that aren't direct descendants of Hex (i.e. Shannon switching game)
 * 2. if there are fewer than three online descriptions of the game
 * 3. games not structurally similar (i.e. not a maker-breaker positional game, not planar, etc)

Three, possibly four, of the items currently on the list would be excluded applying these criteria. What I suggest for these, is to provide a link in the external links section to gamesite(s) where hex variants can be found for gamers (when THAT section overflows with gaming website links, we'll have a separate issue).

Sbalfour (talk) 19:13, 18 January 2017 (UTC)

Embellish article with images
The article has only 3 images, and two of them are dubious: the one illustrating a winning chain doesn't look anything like a chain in an actual game, and may be deceiving (the distribution of stones isn't like that, and Hex is seldom played to the state of a complete chain, just as chess is seldom played to checkmate). The picture of hex on a go board doesn't depict a standard or common playing surface (how about a picture of a 1947 Con-Tac-Tix board, or a 1952 Parker Brothers Hex game board and pieces?)

I find it challenging to imagine what the pattern configurations for path, template and especially ladder look like without a picture, and I'm a hex player. It's like trying to play chess blindfold. I also think we need one or more examples of how these patterns are combined into chains. A few 3 or 4 move opening patterns to illustrate potential connection to templates would also be an appropo addition.

It might also be useful to illustrate how the game develops on 2x2 thru 5x5 boards, because the utility of the patterns is immediately apparent there. This gets to be more of a "magazine" type presentation rather than a scholarly one, so need to keep it concise.

It also seems almost essential to illustrate the asymmetry of the pairwise 2-connectivity paradigm with an obtuse position showing how a play in one area of the board may result in a response in another seemingly unrelated area.

Sbalfour (talk) 17:33, 23 January 2017 (UTC)

Some history of Hex
Sbalfour added a tag noting that statements about the game being popularized in 1942 Danish newspapers, and being manufactured for sale by Piet Hein, really did need some sources. I added a source, from the book "Hex Strategy", about the 1942 newspaper article that first described the game. That book also mentions that Piet Hein marketed the game in 1968, but does not mention any company associated with that marketing, which I think Sbalfour wanted. The book's source for this is a dead Internet link, but that link is still available on Internet archives. That link also fails to mention any company (I wonder if Hein just marketed it himself?). If anyone wants to look at that link to see if there's more information that should be absorbed into this article, it's at: http://web.archive.org/web/20020214224326/http://members.iex.net/~rfinn/gameshlf/abstract/hex/hex.htm. Darrah (talk) 01:43, 26 January 2017 (UTC)


 * I just now mailed the company marketing Con-Tac-Tix and other Piet Hein-related products today, . They say it was manufactured back then by a company called Skjøde Knudsen, or more precisely, Th. Skjøde Knudsen, Skjern. (Th. is short for a first name like Thorvald or Thomas; Skjøde Knudsen a family name; and Skjern, Denmark is the town where the company was situated.) With this information, I've googled the following pages:
 * Boardgamegeek
 * Soma fan page (in Danish and English). It does not mention Con-Tac-Tix, but other wooden Piet Hein products.
 * Hasbro (PDF) Scan of what appears to be original rule book saying "1968", "SKJØDE" etc.
 * So, I have little doubt this is the correct answer, but I don't know if any of these may be considered a valid source. I'll not add anything right now, but knowing the answer may make it eassier to find a source!--Nø (talk) 09:49, 26 January 2017 (UTC)

Online hex 9x9 winning strategy demonstration
I discovered a website that contains an interactive hex game where the computer goes first and always wins. Here it is: http://ec2-34-227-158-207.compute-1.amazonaws.com/fldb/hex99.html

How can this be implemented and is it necessary? I felt like it could be an important resource on this article --IForgot321 (talk) 03:28, 27 January 2019 (UTC)

To always win when moving first on 9x9 without the pie rule is no longer particularly interesting compared to more recent developments in Hex software. There is a bot "Leela" on Little Golem which has been beating up humans on a 13x13 grid, and that includes the pie rule. I would give you a link, but when I did that elsewhere on this same page, my links were removed with the admonishment "Wikipedia is not a gaming site." But I can at least link you to a pdf document which provides a "swap map" for 9x9.

Also if you want to post questions on the Little Golem forum, you must first complete 10 games there I think, or contact me if you like. --Twixter (talk) 18:15, 2 April 2019 (UTC)

Another "Square Hex" Variant
Seems that another "square hex" variant was independently discovered by at least two different people, where the intersections can go both ways. Each player "claims" an intersection if it connects to a space they already moved to. Should we include it? 1) http://hexandoct.net16.net/ 2) https://groups.google.com/d/msg/rec.games.abstract/dhDKiR1edBk/JqIAz9wZaakJ

The "Hex theorem" is ... what?
Under the section Theory and proofs, this passage appears:

"John Nash was the first to prove (c. 1949) that Hex cannot end in a draw, a non-trivial result colloquially called the "Hex theorem", which we now know is equivalent to the Brouwer fixed-point theorem"

Since this does not state what is meant by the "Hex theorem", we can only guess. But readers should not have to guess.

Does this refer to the impossibility of two winners? The impossibility of zero winners? Or both? All three have been referred to by mathematicians as the "Hex theorem".2600:1700:E1C0:F340:597F:5057:E89D:168E (talk) 00:54, 30 April 2019 (UTC)

diagram2
"One such pattern, analogous to the bridge, is shown in diagram 2. " Teheres no diagram 2 in the article. — Preceding unsigned comment added by 89.134.199.32 (talk) 20:13, 15 September 2019 (UTC) 89.134.199.32 (talk) 20:15, 15 September 2019 (UTC).

I came here to say the same thing. The diagrams refer to diagrams in the linked book, I guess. Abd.nh (talk) 19:31, 25 June 2020 (UTC)

is this encyclopedical?
"The game has deep strategy, sharp tactics and a profound mathematical underpinning" this sound more like a marketing slogen to increase apetite to buy this game than a factual desription of the game. 89.134.199.32 (talk) 20:15, 15 September 2019 (UTC)

what the heck is "ala"
"Once placed, stones are not moved, captured or removed from the board (ala, the game of go)" this sentence seems wrong. case 1: if "ala" is supposed to mean "like" or "similarly to" then it is just unnecessary verbosity, since this game is not othervise likened to the game of go and the first part of the sentence already desribes the game clearly enough, so no need for a clarifying remark that does not add to the clarity. case 2: if "ala" is supposed to mean "not like" or "in contrary" then it is just plain wrong since in go pieces need not to be removed, in practice are not removed but instead they become "dead" which means they become owned by the encircling opponent and thus count for the final score of the game. 89.134.199.32 (talk) 20:38, 15 September 2019 (UTC).

Illustration
Neither of the two pictures in the article showing game positions show anything like a realistic position between skilled players. Can we amend that?--Nø (talk) 17:07, 25 October 2019 (UTC)

Wiki Education assignment: Topicos en Ciencias Matematicas
— Assignment last updated by Dlopezramos (talk) 19:41, 22 February 2023 (UTC)

Conjecture about winning moves
At the end of the section Computed strategies for smaller boards, we state a conjecture that the center cell (2 in the case of n being even) is a winning move, uncited. I can find the conjecture in a reference; in fact, the assertion that any play on the short diagonal is a winning move appeared as early as Gardner's 1959 columns in Scientific American. The set of winning moves is in fact considerably larger than that for boards except n=2 and n=4. The number is between 1/2 and 2/3s of nxn. Figuratively, a random first move is most likely to be a winning move! The pattern of likely winning first moves is a critical component of swapping strategy - for very strong automata like we have now, if the first player makes a likely winning move, the second player will swap; otherwise it was a losing move, so the second player must have a winning move, and therefore plays it. So swap patterns, not merely conjectures about a few selected moves, would be an interesting addition to the article. Sbalfour (talk) 17:28, 20 June 2023 (UTC) Sbalfour (talk) 17:28, 20 June 2023 (UTC)
 * I've adjusted this to say what we do know, based on solved board sizes, and cited Gardner's original stronger conjecture. Sbalfour (talk) 19:55, 20 June 2023 (UTC)

Parker Brothers first edition "The Zig-zag Game Hex" copyright 1950
I have a first edition game, and the instruction manual says copyright 1950. I believe we should change the (uncited) date in the article to the copyright date, presumptively the date of first reease. Sbalfour (talk) 20:43, 20 June 2023 (UTC)

Strategy section, redux
The Strategy section is a little anemic: assuming we create on the board various templates with virtual connections, what are we supposed to do with them - i.e. how does that translate into a win? Maybe it's patently obvious, that we must attempt to create an unbroken chain of templates and/or stones from one edge to the other?! Maybe that's why we don't say it. But it'd complete the argument that templates are useful to do so. If we do say that, is it necessary to have a source? If it's patently obvious so we don't say it, then it should be patently obvious when we do, i.e. no source needed. I'm not sure I entirely buy my own argument; I believe we should be able to find such a source that says it, or generally implies it. Sbalfour (talk) 20:02, 21 June 2023 (UTC)

I notice that editor selinger on 19 July, 2022 replaced the previous Strategy section wholesale with no explanation (see this edit). The old section relied exclusively on a single source (Brown), mostly because only one book has ever been published on the game. And page numbers were missing from most of the refs. Earlier, I myself replaced the Strategy section wholesale with this edit, for reference to where we've been. Sbalfour (talk) 18:30, 23 June 2023 (UTC)

There are other issues with the new section. If we are going to use virtual connection for strongly connected stones, we almost certainly need to define virtual semi-connection for weakly connected stones. They are complementary kinds of connectedness, and both equally essential to understand. Gardner's original article on the game clearly described (and diagrammed) the geometry of connectedness without jargon. The section contains game theory jargon MOS:JARGON: template, internal template, external template, virtual connection, carrier as well as game playing jargon: bridge, trapezoid, span, crescent, A2, A3, D4, A4. These are not widely used terms even in the limited literature we have. And I question whether the diagrams of specific geometric patterns is WP:GAMEGUIDE. Sbalfour (talk) 17:46, 23 June 2023 (UTC)

A decent Strategy section should provide a simple answer to such questions as: I should emphasize that this section (and the whole article) are about the strategy (game), not how to play it. Sbalfour (talk) 18:03, 23 June 2023 (UTC)
 * How does one start the same (specifically, how to treat the swap rule)
 * What does one do after that? (general strategy)
 * What are useful geometric motifs (tactics)
 * What does the end of the game plausibly look like (objective). This is analogous to chess where the objective is checkmate, but the actual end of the game is hardly ever checkmate.

Hi, I just saw the changes you all made to the strategy section. To be sure, the edits I made in July 2022 were intended to be the beginnings of a useful strategy section, not the entire section. I wrote a section on templates, hoping that other similar sections could be added later (say on ladders and ladder escapes, and other strategy topics). As for sources: Brown's book was published in 2000, as was Anshelevich's article, and both must be considered unreliable sources according to WP:RS, because they are old and outdated. Most of what was known about Hex strategy in 2000 was been superseded. On the other hand Seymour's book, although self-published, can be considered a reliable source. His book has been widely read by other Hex players and is currently the most up-to-date and comprehensive source on Hex strategy. Seymour himself is a top Hex player, currently ranked 5th in 13x13 on LittleGolem.net, where most of the world's top players play. Although his book is self-published, it has been cited by published academic sources (for example G3 of http://math.colgate.edu/~integers/vol22.html, of which I am the author, but which was decidedly refereed and not self-published). For these reasons, and also because no better sources are available, I believe Seymour's book should be admissible as a reliable source on Hex strategy.

I agree with the criticism of my original July 2022 edits (e.g., sources should have been added), but I think what it has been replaced with is far worse. Selinger (talk) 01:23, 11 January 2024 (UTC)


 * Additionally, I just saw that Seymour's book is also cited in Hayward, Ryan B. Hex: A Playful Introduction. Vol. 54 of Anneli Lax New Mathematical Library. American Mathematical Society, 2022. So now there are at least two independent reputable published sources who cite Seymour's self-published book as an authority on Hex strategy. I think this strengthens my argument that going forward, Seymour's book is not merely a blog. It is a source that other reliable authors rely on. WP:RS does not prohibit self-published material outright. It merely establishes a higher standard for assessing their reliability. Selinger (talk) 01:47, 11 January 2024 (UTC)

About a source
The Strategy section is not adequately sourced even if its sole source were valid. That source was. That source is by a non-credentialed author, self-published, online only, and not peer reviewed. The content basically appears on a personal blog which is not allowed in Wikipedia. I have therefore removed it leaving the whole section unsourced. Unsourced content may be challenged and removed, and I have challenged it with an unsourced tag. In due time, if WP:RS sourcing does not appear, I will remove the section and revert it back to the previous rendition of 19 July, 2022. That rendition has issues also, but they were more repairable than the ones faced here. Sbalfour (talk) 14:51, 26 June 2023 (UTC)

First player win, informal existence proof
We state the following: "This extra piece cannot interfere with the first player's imitation of the winning strategy, for an extra piece is always an asset and never a handicap." We're all nodding our heads and moving right along, stepping smartly. But is that so? Why? Fundamentally, it's because hex is a monotone game. Ahhh, but we haven't proven that yet. We haven't even said that yet. And proving it is a piece of work. Sbalfour (talk) 05:31, 6 July 2023 (UTC)


 * Well, I'd argue it's a very intuitive fact which is also correct, which in the context of Wikipedia does not strictly need a proof. But in any case, that statement does not fit well in the argument anyway: What is needed here is merely that final positions in Hex are monotone, i.e., the extra piece cannot change a won position for the first player to a non-won position (for the first player), nor can it change a non-won position for the second player to a won position (for the second player). The rest of that section is already proving that monotonicity of final positions implies monotocity of all positions with the strategy imitation argument.
 * I'd like to see if there is some way to phrase this property better before editing the article, though. Bbbbbbbbba (talk) 11:26, 11 July 2023 (UTC)