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?

Two players alternatively erase some 9 numbers from the sequence 1,2,...,101
until only two remain. The player that starts wins x−54 dollars from the
player that plays second, x being the absolute value of the difference between the remaining
two numbers.

Would you rather be the first or the second player?

Explain your decision by providing your strategy.

Andy rolls five regular dice.

What is more likely: rolling no sixes or rolling exactly one six?