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?
(In reply to re: my solution
by Ady TZIDON)
thank you for point this out. I had used an Excel table to evaluate all the possible combinations and it seems there was a flaw in my design of this table.
So to modify my previous analysis,
let us assume that at the start of the round the final digit is x, and Erdos chooses a number with a final digit of y
Then for any value of x other than 0 and 5, Oleg is able to make a choice that prevents the final digit from being 0. The inclusion of 5 here is what I missed in my previous analysis but it still seems that my logic still stands with a slight modification. It seems now that Oleg needs to prevent the final digit from being both 0 or 5. From what I am seeing this is also possible.
For any value of x other than 0 or 5, then Oleg can make a choice that prevents the new value from being either 0 or 5 mod 10 regardless of what Erdos chooses.
At this point I am at a loss as to where my logic has failed so I will leave it to someone else to possibly discover while I think it over, I will include my logic table in a separate post to further show my reasoning.
Posted by Daniel
on 2017-05-27 09:49:29