Question 50
Francesca and Giorgia play the following game. At the start there are some columns of coins on a table. Each column contains a certain number of coins, which may differ from column to column. Taking turns, each player makes one and only one of the following possible moves:
- she chooses a column containing a nonzero even number $2k$ of coins and replaces it with two columns containing $k$ coins each;
- she removes from the table all the columns containing an odd number of coins.
If a move of the first type is not possible, the player must make one of the second type, and vice versa.
Francesca starts. The player who takes the last coin off the table wins.
a. If initially there is a single column on the table, containing $2008^{2008}$ coins, which player has a winning strategy?
b. For which initial configurations does Francesca have a winning strategy?
- she chooses a column containing a nonzero even number $2k$ of coins and replaces it with two columns containing $k$ coins each;
- she removes from the table all the columns containing an odd number of coins.
If a move of the first type is not possible, the player must make one of the second type, and vice versa.
Francesca starts. The player who takes the last coin off the table wins.
a. If initially there is a single column on the table, containing $2008^{2008}$ coins, which player has a winning strategy?
b. For which initial configurations does Francesca have a winning strategy?