New Stuff
An 8x8 opening strategy.
Hex Theory and Proofs Project Main Page
This is just a beginning.
Besides the material here, there is a Google Group specifically
for the theory of the game. You can access the archives here:
To receive emails or to contribute you must become a member, but anyone can browse the archives.
Current Events
Well, this page is not going to be maintained well enough to keep up with everything, so
I'm going to point at the two most prolific researchers of the hex game that I know of. They've published together and independently, covering a lot of ground.
 Ryan Hayward, Professor at University of Alberta publications
 Javhar, (Jack van Rijswijck) publications
Formal Theory
The purpose of the Theory and Proofs Project is to collect in one
place proofs of all of the major theoretical results concerning the
game. If we also wind up proving something new, so much the better.
It is our intent to generate sound and rigorous, albeit informal
proofs of a number of known facts which have been proved elsewhere,
but in sources that are diverse in accessability, location, and
treatment. These include but are not limited to the following:
 Lemmas:
 hex game cannot end in a draw.
 If one player wins, the other player does not win (a bit stronger than Lemma 1).
 A hex game, if play continues
long enough, ends in a win for one or the other player (a bit
stronger than either of the previous Lemmas.)
 Either the first or the second
player has a winning strategy on the starting (empty) position.
 If a (winning) strategy exists
for player A in a given position, a strategy exists for that player
in any derived position that differs from the original only in
having additional plays by the winning player
 If a (winning) strategy exists
for player A in a given position, a strategy exists for that player
in any derived position that differs from the original only in the
deletion of some plays by the losing player
 In every finished game (where
some side has actually won), the winning path can be reduced to a
minimal winning path in which no hex borders more than two other
hexes, and the goal edges are touched by exactly one hex apiece
from the minimal winning path.
 Theorems:
 In the absense of a "swap"
rule, if the second player has a winning strategy for the starting
position, then the first player has a winning strategy for the
(same) starting position, contradicting Lemma 2.
 In the presence of a "swap"
rule, it is the second player who has a winning strategy.
 A1 is a losing opening. Proven in
Excursions into Mathematics (original 1969 edition, also appears in
Millenium Edition).
 A2 is a losing reply to A1.
Proven in Excursions
into Mathematics: The Millenium Edition.
 B1 is a losing opening. There is a sketch
of the proof.
Library Section
This collection of documents contains some proofs of the items
listed above, as well as other things. It is intended to be a
bibliography and direct resource for anyone interested in the
theories behind hex, computer programs that play hex, and game
analysis in general.
Some documents here are restricted in distribution. Accordingly,
they are available only by prior arrangement. You can get access by
contacting the Hex
Master, indicating your real name and a useable email address,
and a desired username (none of which will be used, sold, published,
or listed in any other place.) On approval, you will have access to
the restricted material here, as well as the latest test
build of the OHex database.
Other documents were found on the web or elsewhere with a more
permissive copyright notice, or no notice at all; these are readable
here without special permission.
Scanned pages (sorry, may take a while to download):
 Hex
and the Brouwer FixedPoint Therem
 Winning
Ways for your Mathematical Plays (second edition).
 A
Combinatorial Problem Which is Complete in Polynomial Space.
S.Even and R.E.Tarjan  Journal of the Association for Computing
Machinery, 1976 23(4): 710719.
 Four artices from Proceedings
of Computers and Games Conference, 1998
 Three artices from Proceedings
of Computers and Games Conference, 2000
 Five articles from Proceedings
of Computers and Games Conference, 2002
 Excursions
into Mathematics: The Millenium Edition. A. Beck, M.N.Bleicher
and D.W.Crowe, eds, A.K.Peters, Natick, MA.
 Six
Wins Hex Tournament. Gabor Melis and Ryan Hayward  ICGA
Journal, 2003 26(4): cover, 278280. Part of a report on the 8th
Computer Olympiad, covering the hex competition between Six and
Mongoose.
 Hex is PSPACEComplete,
Stefan Reisch, Acta Informatica 15, 167191 (1981).
This is the original paper, in German. An English translation
is listed below.
 UnionConnections and Straightforward Winning Strategies in Hex
Kohei Noshita – ICGA Journal, 2005 28(1): cover, 312. A winning strategy for 8x8 hex.
Various formal papers:
YLP01 Yang, Liao, Pawlak,
2001, A decomposition method for
finding solutions in game Hex 7x7, International Conference on
Application and Development of Computer Games in 21st Century.
 Ans02 PDF
Anshelevich, V. (2002) A Heirarchical Approach to Computer Hex.
Arificial Intelligence, 134(12): 101120.
There is also a slide presentation.
 RIJ02 PDF
Rijswijck, J. v.(2002). Search and Evaluation in Hex. Technical
report. University of Alberta.
 HBJ02 PDF
R. Hayward, Y. Björnsson, M. Johanson, M. Kan, N. Po and J. van
Rijswijck. Solving 7x7 Hex: Virtual Connections and Game State
Reduction Advances in Computer Games 10, Graz, 2003. The
underlying data is also available on
the web.
 YLP02 Yang, Liao, Pawlak,
2002, A new solutions for 7x7 hex
game, Proceedings of Computers and Games Conference, 2002. There
is a corresponding technical report.
 Hay02 Berge and the Art of Hex,
a chapter for a book, recounting some history and theory surrounding
Hex. Interestingly, "Excursions in Mathematics" is
credited with proving B1 is a losing opening, but the book itself
does not have that proof, but the slightly weaker one that A2 is a
losing reply to A1.
 REI81 PDF
S. Reisch. Hex is PSPACEComplete
T. Ace translator. From Acta Informatica 15, 167191 (1981).
A formal proof of the complexity of generalized Hex. Original
German version available above.
Additional data files:

Some years ago, Kevin Walker
(kevin at canyon23 dot net) exposed an applet version of Jhex on his web
page, with the intent that the community of players could contribute
to solving 7x7 hex. The results of some months of work by some of
the best players of the time is here (click
to download). It was wrong in a number of respects, largely
because of winning moves that were not even tried. Fortunately,
there is no record of the guilty parties, although I'll admit to
having worked out some of the variations on A2 and A3 –
probably all of them wrong.
Corrections are not posted
because they are the topic of an ongoing set of puzzles in the OHex
mailing list.
Last updated 9 June 2005. Minor edits 26 Dec 2013.