5 ms·

how is it guaranteed that games do not not end in a draw?

• tromp 5y ago
  The hexagonal diamond board topology is such that an east-west connection can only be prevented by a north-south connection. For more formal proofs, see https://sites.google.com/site/javhar1/hexcannotendinadraw https://sites.google.com/site/javhar1/hexcannotendinadraw
  • bo1024 5y ago
    Nice diagrams and proofs! This is one of those that can feel obvious, but needs a bit of deep math
• paxys 5y ago
  If you draw a hex board and try out some connections it will become immediately clear. There can never be a deadlock. No matter how you fill out all the tiles, either the white OR the black sides will connect.
• karmakaze 5y ago
  A square arrangement if we allow 8-way connections, we can end up with a cross-connection: ox xo If we only allow 4-way connections then a connected line won't permit another line to cross it. For hex 6-way connections, lets say a place ( ) is where lines cross, but it can only be a single color. How about making a cross connection like the 8-way square: (doesn't work) x x x o x o o o x x o x x x o x x x
  • shagie 5y ago
    If you put the constraint that an 'x' set of connections on a square board is not allowed - ox x. So that 'o' is prohibited from playing in the empty spot there because 'x' already has a connection - you get the game Crossway - https://boardgamegeek.com/boardgame/30517/crossway https://boardgamegeek.com/boardgame/30517/crossway // http://superdupergames.org/gameinfo.html?game=cross http://superdupergames.org/gameinfo.html?game=cross // http://www.marksteeregames.com/Crossway_rules.pdf http://www.marksteeregames.com/Crossway_rules.pdf
• kevinwang 5y ago
  My attempt at intuition: the very act of completely blocking your opponent from making a connection will make a connection for yourself.
• Syzygies 5y ago
  Wow. That takes me back. John Nash independently invented hex in grad school. A sequence of scenes involving hex were cut from the movie "A Beautiful Mind". The first day that Russell Crowe joined the production, he was occupied with hours of makeup tests for his aging progression. Ron Howard had me meet him during a break, to discuss the hex and go scenes to come. The first dozen of a crew that would be hundreds hadn't met him either, so they hovered nearby as we talked. To my surprise, Russell asked me what mathematical truths drew Nash to this game. Never underestimate people; I went for it. Rather than explaining the proof Nash liked that the first player could always win, I decided to explain why some player had to win. As I talked, Russell drew me in with fascinated, understanding facial expressions. When I was done he looked at Ron, "Can you get me the Parker Brothers rules? They're for 8 to 80, I didn't understand a word this guy just said!" Peals of laughter from the room. Months later we're filming a roomful of grad students playing hex, and Russell doesn't like what the script has him saying. He gets permission to ad lib. "Think of the black stones as water, the white stones as land. Either you can swim across this way, or there's a land bridge that way..." My longest day shooting, I woke at five, and was an actor in the pen ceremony that evening. Close to 24 hours awake, I found myself nodding off high at a party in Russell's room, everyone singing various national anthems. I learned much later that Russell had been kidded as a child for not knowing the New Zealand anthem. By the party he knew them all. He likes jokes only he gets, jokes with long fuses.