Name:      

Directions: Show all work.

1. In a game of Nim, tokens are placed in various piles. In each turn, players removing one or more tokens from a selected pile. In this variant, the game begins with $n$ piles, each starting with 10 tokens. When no tokens remain, the player to move loses.

   1. [3 points] Prove that if $n$ is even, then Player 2 has a winning strategy. (Hint: Analyzing the general game of Nim is tricky; give an explicit strategy for Player 2 in this special case.)
   2. [2 points] Use part (a) to show that if $n$ is odd, then Player 1 has a winning strategy.

2. [5 points] Let $a_0=0$ and $a_n=\frac{1}{2-a_{n-1}}$ for $n\ge 1$. Guess a formula for $a_n$ and prove your formula is correct.