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One persona may not necessarily have the same knowledge of the other personas.
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In The shadow of the past, FotR, Gandalf tells about Gollum in the Misty Mountains that "there was nothing more to find out, nothing worth doing, only nasty furtive eating and _resentful remembering_". Anyway, a Gollum without the knowledge possesed by Smeagol would make a poor survivor, even in the wilds, however small that knowledge may be.
Concerning luck and cheating, I think that the following passages are also relevant(from the Prologue):
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In the end Bilbo won the game, more by luck (as it seemed) than by wits; for he was stumped at last for a riddle to ask, and cried out, as his hand came upon the ring he lad picked up and forgotten: _What haw I got in my pocket?_ This Gollum failed to answer, though he demanded three guesses. The Authorities, it is true, differ whether this last question was a mere 'question' and not a 'riddle' according to the strict rules of the Game; but all agree that, after accepting it and trying to guess the answer, Gollum was bound by his promise.
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