Gollum inadvertently showed him the way out because he thought ''It found the way in, it must know the way out!'' So he ran to the exit in order to catch him, but he was following him the whole time.
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Bilbo used his new ring to stay invisible and was able to follow Gollum to the exit. He had to leap over Gollum and charge through the door. His buttons got stuck and he had to leave them behind to get away. The orcs had seen his shadow and he had no time to extricate himself before they got him. So he just pushed through and lost the buttons.
Bilbo said, talking to himself, "what have i got in my pocket?" (it was the Ring). Gollum thought it was a riddle and Bilbo decided to make it one. he gave Gollum three chances to guess and gollum lost after four. then gollum went "to get something" (it was the ring), and Bilbo thought he was abandoned and left with the ring on. Gollum thought Bilbo knew the way out and he wanted to follow him, realizing Bilbo had the ring. Bilbo saw Gollum racing towards the gate and followed him to safety.
Bilbo found an old, worn stairway.
Balin goes part of the way down the tunnel with Bilbo, and Bilbo takes one single golden goblet from Smaug’s immense hoards.
Ah, what a lovely question! "Between the flying rags" in The Hobbit refers to the moment when Bilbo is trying to escape the goblins in the Misty Mountains. It's a whimsical way of describing the chaos and confusion of the scene, where Bilbo is darting between the goblins' legs and cloaks, trying to find his way out. Just imagine the swirling colors and movement in that scene, like a dance of bravery and quick thinking!
The dwarves left a note for Bilbo under the mantle clock. It stated that they would be awaiting him at 11 am sharp. Bilbo didn't get the note until 10 minutes before that and ran the whole way to be on time.