Question 107
Consider a row of numbered boxes $0,1, \ldots, k$ from left to right where, for each $i \geqslant 1$, box number $i$ holds $x_{i}$ tokens. There is initially no tokens on box number $0$. In turn, Alice and Bob then play according to the following rules: - Bob chooses a set $S$ of tokens, not necessarily all on the same box. - Alice can then either eliminate all tokens that are not in $S$ But then move each token from $S$ of the box he occupies at the neighboring box on his left (such a token goes from a box number $i$ In box number $i-1$), or eliminate all tokens that are in $S$ But then move each token that is not in $S$ of the box he occupies at the neighboring square on his left. Bob wins the game if he manages to bring a token to box number $0$, and Alice wins if she manages to eliminate all the tokens. 1) Prove that Alice has a winning strategy if $\sum_{i=1}^{k} 2^{-i} x_{i}\lt 1$. 2) Is it true that if $\sum_{i=1}^{k} 2^{-i} x_{i} \geqslant 1$ So Bob has a winning strategy?