If Rita goes first in a stick-removal game, which values of n guarantees Sam can always win?

In a stick-removal game, players take turns removing one or more sticks from a pile, and the player who removes the last stick wins. The strategy involves understanding winning and losing positions, particularly what numbers of sticks guarantee that one player can force a win regardless of the opponent's moves.

When analyzing the game, a player has a winning strategy if they can move the game into a losing position for their opponent. A losing position is one from which any move will leave the opponent in a winning position.

In this context, we can determine specific values of n (the number of sticks) that guarantee a win for Sam when Rita goes first. Here's how the analysis works for the choices given:

• For n = 1, 2, 3, 4, and 5, Rita can always take the last stick, leaving Sam with no options to win.

• For n = 6, no matter how many sticks Rita removes (1 to 5), she will leave Sam with a number of sticks (1 to 5), from which he can always win on his subsequent turns.

When Rita goes first with 6 sticks, Sam has a guaranteed strategy to win. If Rita removes 1 stick, Sam will have