firstname.lastname@example.org (Bill Taylor) wrote in message news:<email@example.com>... > It is an old theorem that in Hex, once the board has been completely > filled in with two colours, there *must* be a winning path for one > or other of them. > > Now, I can prove this easily enough mathematically, but I'm wondering if > there is a simple proof, or proof outline, that would be understandable > and reasonably convincing to the intelligent layman. > > Can anyone help out please?
Imagine the board tipped up, so player A goes from top to bottom, player B from left to right. Suppose the board is actually two sheets of clear plastic, and two players have "stones" of white sugar (A) and red granite (B), and moreover these stones "plug" the hexagon cells (in the obvious way).
Now pour water in at the top. Either it dissolves through the sugar and reaches the bottom, in which case A has a path, or it doesn't, in which case there must be a continuous boundary of impervious granite, and B has a path.
(Does this help? I think it does (slightly) use some facts about the behaviour of liquids to help the imagination along.)