is not difficult to prove that Hex cannot end in a draw, 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. Hein also stated this fact as "a barrier for your opponent is a connection for you". John Nash wrote up a proof of this fact around 1949, but apparently didn't publish the proof. Its first exposition appears in an in-house technical report in 1952, 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. In 1979, David Gale published a proof which also showed that it can be used to prove the two-dimensional Brouwer fixed-point theorem, and that the determinacy of higher-dimensional variants proves the fixed-point theorem in general. A 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 doesn't, 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. First-player winning strategy 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.". 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. The proof works for a number of games including Hex, and has come to be called the strategy-stealing argument. Since it