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.

It also has profound mathematical underpinnings related to the Brouwer fixed-point theorem, matroids and graph connectivity.

[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.

"[1]: 134  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.

[7] 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.

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

This program was based on Polygames[13] (an open-source project, initially developed by Facebook Artificial Intelligence Research and several universities[14]) using a mix of:[15] 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.

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.

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.

This fact was known to Piet Hein in 1942, who mentioned it as one of his design criteria for Hex in the original Politiken article.

Its first exposition appears in an in-house technical report in 1952,[24] in which Nash states that "connection and blocking the opponent are equivalent acts".

[26] 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.

In Hex without the swap rule on any board of size nxn, 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".

[27] A strengthening of this result was proved by Reisch by reducing the quantified Boolean formula problem in conjunctive normal form to Hex.

[33] 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.

[35] 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.

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.

According to the book A Beautiful Mind, John Nash (one of the game's inventors) advocated 14×14 as the optimal size.

The misère variant of Hex is called "Rex", in which each player tries to force their opponent to make a chain.

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.

Projex is a variation of Hex played on a real projective plane, where the players have the goal of creating a noncontractible loop.

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.

[citation needed] One of the largest Hex competitions is organized by the International Committee of Mathematical Games in Paris, France, which is annually held since 2013.