A game with 330 pebbles

Question

Anastasia and Barnabas play a game that starts with $330$ pebbles in a bowl. The game consists of two phases. The first phase looks as follows:

• First Anastasia announces an integer $A$ with $2\le A\le9$.
• Then Barnabas announces an integer $B$ with $2\le B\le9$ and $B\ne A$.

The second phase looks as follows:

• The two players alternately take pebbles out of the bowl. Anastasia makes the first move.
• In every move, Anastasia may either take $1$ pebble or $A$ pebbles.
• In every move, Barnabas may either take $1$ pebble or $B$ pebbles.
• The player who takes the last pebble wins the game.

Question: Which player is going to win this game? (As usual, we assume that Anastasia and Barnabas both use optimal strategies.)

Answer 1

Answer

Anastasia wins

Because

Anastasia is the first to choose the integer and she chooses $A = 2$.

Anastasia should first reach a status where there's a number of pebbles left less than $B$.
To achieve this, on every Anastasia's turn we can say that there are

$B + n$ pebbles left

As long as $n > 2$, she can play whatever number. When $n \leq 2$ Anastasia must play the correct number: if $n = 2$ she plays $1$, else if $n = 1$ she plays $2$.
The status after this move is either $B + 1$ or $B - 1$ and it's Barnabas' turn. He cannot win at this turn (because $B \geq 3$) and after his turn we have less than $B$ pebbles left.
Anastasia should now just make sure that after her turn there's an even number of pebbles left, so that Barnabas' can't win. On the last turn there's just 1 pebble left and it's Anastasia's turn.