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?
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.