Problem E

Taking Grapes

Quinn and Birgitta are playing a game using a bowl of grapes. The rules of the game are as follows:

Write the numbers \(1\), \(2\), \(3\), \(4\), and \(5\) on a whiteboard.

Players take turns choosing a number from the whiteboard, removing that number of grapes from the bowl, and then erasing that number from the whiteboard.

The game continues until all the numbers are erased or a player is not able to take any of the remaining numbers of grapes from the bowl.

The last player who erased a number from the whiteboard is the winner.

The game starts with \(8\) grapes and Quinn goes first. Quinn has developed a winning strategy so that she is guaranteed to win this game, regardless of how many grapes Birgitta takes on her turns. Find all the possible first moves in Quinnâ€™s winning strategy. Justify your answer.