Editing Hex (board game) (section)

===Determinacy===
It is not difficult to convince oneself by exposition, that hex cannot end in a draw, referred to as the "hex theorem".  I.e.,  no matter how the board is filled with stones, there will always be one and only one player who has connected their edges. This fact was known to Piet Hein in 1942, who mentioned it as one of his design criteria for Hex in the original Politiken article.<ref name="HexFullStory"/>{{rp|29}}
Hein also stated this fact as "a barrier for your opponent is a
connection for you".<ref name="HexFullStory"/>{{rp|35}} John Nash wrote up a proof of this fact around 1949,<ref>{{cite journal |last1=Hayward |first1=Ryan B. |last2=Rijswijck, van |first2=Jack |title=Hex and combinatorics |journal=Discrete Mathematics |date=6 October 2006 |volume=306 |issue=19–20 |pages=2515–2528 |doi=10.1016/j.disc.2006.01.029 |doi-access= }}</ref> but apparently did not publish the proof.  Its first exposition appears in an in-house technical report in 1952,<ref>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 {{webarchive |url=https://web.archive.org/web/20170121094752/https://www.rand.org/content/dam/rand/pubs/documents/2015/D1164.pdf |date=21 January 2017 }}</ref> in which Nash states that "connection and blocking the opponent are equivalent acts". A more rigorous proof was published by [[John R. Pierce]] in his 1961 book ''Symbols, Signals, and Noise''.<ref>{{cite book |last1=Hayward |first1=Ryan B. |last2=Toft |first2=Bjarne |title=Hex, inside and out : the full story |year=2019 |publisher=CRC Press |location=Boca Raton, Florida |isbn=978-0367144258 |page=99}}</ref> In 1979, [[David Gale]] published a proof that the determinacy of Hex is equivalent to the two-dimensional [[Brouwer fixed-point theorem]], and that the determinacy of higher-dimensional ''n''-player variants proves the fixed-point theorem in general.<ref>{{cite journal|author=David Gale |year=1979|title=The Game of Hex and Brouwer Fixed-Point Theorem | journal=The American Mathematical Monthly | volume=86 | pages=818–827|doi=10.2307/2320146|jstor=2320146|issue=10|publisher=Mathematical Association of America}}</ref> 

An informal proof of the no-draw property of Hex can be sketched as follows: consider the connected component of one of the red edges. This component either includes the opposite red edge, in which case Red has a connection, or else it does not, in which case the blue stones along the boundary of the connected component form a winning path for Blue.  The concept of a connected component is well-defined because in a hexagonal grid, two cells can only meet in an edge or not at all; it is not possible for cells to overlap in a single point.