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 a_{i} to Erdös, who then chooses a non negative integer b_{i}, and then Oleg defines a_{i+1} = |a_{i}-b_{i}| or a_{i+1} = a_{i} + b_{i}.

Erdös wins if a_{20} is a power of 10, otherwise Oleg wins.

Who is the winner, Oleg or Erdös?

In

"5 dice" Andy had five regular dice. Now he has a total of N regular dice. He claims that the odds of rolling exactly M sixes is exactly half as likely as rolling (M-1) sixes. (M < N).

For what values of N is this true?

State the pattern if there is one.

Express M as a function of N.

Let N(x) be the number 122....221 where the digit 2 occurs x times.

Twice in the

past we have determined the highest power of 11 that divides N(2001) is 11^3.

What is the smallest x for N(x) to be a multiple of 11^3? What about multiples of 11^4 and 11^5?