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 Oleg or Erdös? (Posted on 2017-05-26)
Oleg and (the ghost of) Erdös play the following game. Oleg chooses a non- negative integer a1 with at most 1000 digits.

In Round i the following happens:
Oleg tells the number ai to Erdös, who then chooses a non negative integer bi, and then Oleg defines ai+1 = |ai-bi| or ai+1 = ai + bi.
Erdös wins if a20 is a power of 10, otherwise Oleg wins.

Who is the winner, Oleg or Erdös?

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 Thoughts | Comment 4 of 8 |
Let p_i be the smallest tenth power not exceeding a_i.  Have Erdos chooses b_i = p_i - a_i.  Oleg loses immediately if he adds them (Erdos can then choose 0 afterwards), so he subtracts them.  This does not guarantee a win (the a_i sequence can get stuck in a cycle 6,2,6,2,...), but it does allow Erdos to control what terms are in the a_i sequence.

 Posted by Brian Smith on 2017-05-27 09:56:49

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