MathDB

Abby and Brian play the following game: They first choose a positive integer

N

. Then they write numbers on a blackboard in turn. Abby starts by writing a

1

. Thereafter, when one of them has written the number

n

, the other writes down either

n + 1

2n

, provided that the number is not greater than

N

. The player who writes

N

on the blackboard wins. (a) Determine which player has a winning strategy if

N = 2011

. (b) Find the number of positive integers

N\leqslant2011

for which Brian has a winning strategy.
(This is based on ISL 2004, Problem C5.)

Problem Statement