Bilbo hides the Arkenstone from the dwarves because he knew each one was greedy for the Arkenstone and they might kill each other or harm others for it.
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In chapter 5; Riddles in the dark; bilbo finds a ring.
Smaug wasn't exactly hiding. He was laying in the main chambers in the heart of the mountain. Bilbo just went straight down the tunnel from the "side porch".
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.
Gandalf surprised Bilbo and he was foreshadowing the future
Bilbo's mother was called Belladonna.