B1 is a Losing Opening for V(Sketch of Proof)

This is just a sketch of the proof. As it is written, this site does not have actual proofs of any of the lemmas used. This is posted in the hopes someone cares enough to critique the approach.

This sketch is just a minor restatement of a proof by Holland an Beck in Excursions into Mathematics: The Millenium Edition, where it is shown that A2 is a losing reply to A1. By itself, this fact strongly suggests that B1 is a losing opening, but does not quite accomplish proof of that fact. While it is hard to imagine that the authors of the original proof did not notice that by leaving A1 out of their consideration they have proven B1 a loser, they did not restate their proof or claim that result.

Since every opening is either winning or losing, let us assume that B1 is a winning opening; that is, that there is a winning strategy for the first player that begins with a play at B1. We hope to prove that we can derive a winning strategy for the second player who responds with B2. Then both players will be seen to have a winning strategy in the same position; but by our Lemma 2 this cannot happen, and the contradiction disproves the assumption that B1 wins.

On the assumption that B1 wins for V playing first, and repeated application of Lemma 5, there is a winning strategy for the position in which the opening V player is allowed moves at B1, C1, A2 and B2 before the second player moves at all. We will have the second player win with a modified version of this same strategy. It will arise from the corresponding position where the second player (playing H) has plays at A2, A3, B1 and B2 and the V player is to play with no plays of their own on the board. By the inherent symmetry of the game, this must also be a win for H.

Let game proceed without those extra moves, with a reply at B2, so that the game so far has gone simply 1) B1 B2. Despite the actual dispositon of plays, we have the H player proceed as if all of the hexes A2, A3, B1 and B2 are occupied by H, with these minimal elaborations to accomodate the reality. Note that Hexes B1 and B2 are actually occupied, and that the basic strategy being followed will not call for H to make plays on A2 or A3, because the strategy assumes H already occupies those hexes. There is nothing, however to prevent the V player from taking those locations, however. If this should occur, we specify that a play at A2 should be answered with a reply at A3, and vice-versa. With this addition, the strategy specifiesy replies for any possible moves by V.

Let this strategy be followed until it indicates that the H player has won, and we will find that indeed the H player has won, or has a trivially winnable position despite the false assumption that the positions started with 6 H plays when in fact there was only one.

To see this, consider the presumed winning path, and the minimal winning path promised by Lemma 7. If that path does not go through C1, C2, B3 or A4, then it is in fact a winning path. If however, it goes through one of these, consider these cases:

To do: this sketch's treatment of the minimal path is not quite rigorous. I should show that it touches the corner position in at most one spot. That will be just a variation on the proof of Lemma 7, but as I don't have that yet, it's missing here too.