Editing Hex (board game) (section)

===First-player win, informal existence proof===

In Hex without the [[swap rule]] on any board of size ''n''x''n'', 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".<ref name="HexFullStory"/>{{rp|42}} 

All known proofs of this fact are non-constructive, i.e., the proof gives no indication of what the actual winning strategy is.   Here is a condensed version of a proof that is attributed to John Nash c. 1949.<ref name="Gardner1"/> The proof works for a number of games including Hex, and has come to be called the [[strategy-stealing argument]].

# Since Hex is a finite two-player game with perfect information either the first or second player has a winning strategy, or both can force a draw by [[Zermelo's theorem (game theory)|Zermelo's theorem]].
# Since draws are impossible (see above), we can conclude that either the first or second player has a winning strategy.
# Let us assume that the second player has a winning strategy.
# The first player can now adopt the following strategy.  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.<ref group="note">If the board has been completely filled, then one player must have won already, and it must be the first player since they have been playing a winning strategy.</ref>  In this way they play the winning strategy with one extra piece always on the board.
# This extra piece cannot interfere with the first player's imitation of the winning strategy, for an extra piece is never a disadvantage. Therefore, the first player can win.
# Because we have now contradicted our assumption that there is a winning strategy for the second player, we conclude that there is no winning strategy for the second player.
# Consequently, there must be a winning strategy for the first player.