This question is from TIFR exam.
Consider the following game with two players, Aditi and Bharat. There are $n$ tokens in a bag. The players know $n$, and take turns removing tokens from the bag. In each turn, a player can either remove one token or two tokens. The player that removes the last token from the bag loses. Assume that Aditi always goes first. Further, we say that a player has a winning strategy if she or he can win the game, no matter what the other player does. Which of the following statements is TRUE?
- For $n=3$, Bharat has a winning strategy. For $n=4$, Aditi has a winning strategy.
- For $n=7$, Bharat has a winning strategy. For $n=8$, Aditi has a winning strategy.
- For both $n=3$ and $n=4$, Aditi has a winning strategy.
- For both $n=7$ and $n=8$, Bharat has a winning strategy.
- Bharat never has a winning strategy.
My attempt:
Since, both player knows value of $n$, and Aditi always starts the game. Also both player can choose only either one token or two tokens.
Therefore, I assume that $n=3$, and Aditi picks two, now Bharat can picks only remaining one token from the bag. So, Aditi won the game.
Now assume that $n=4$, and Aditi picks two, now Bharat picks one then Aditi can picks only remaining one token from the bag. So, Bharat won the game. If, also, Aditi picks one, now Bharat picks two then Aditi can picks only remaining one token from the bag. So, Bharat won the game.
Similarly, (alternative) for $n=7$, Bharat has a winning strategy. For $n=8$, Aditi has a winning strategy.
So, I chosen option $(2)$, but one of the my friends chosen, option $(5)$.
Can you explain it, please?