Let's say a blue dwarf makes the final connection to the gold. However, there are two paths that one could take to get from the ladder to the gold. One of these paths has a blue door, one of them has no doors. Obviously the blue dwarf would prefer to use the path with the door, so does the player who makes the final connection get to choose what path to use? Or does it go by shortest path to the gold? Basically, what are the rules to determine what path counts as the "final path"?
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There's no such concept. The following are the win conditions for the red and blue teams from the rules:
A dwarf from the blue team created the connection, and there is a path that's not blocked by a red door, so the blue team wins, and the red team does not. That's it.
The only way the red team can win if a dwarf from the blue team creates the connection is if the blue team can't win (i.e. every path is blocked by red doors).