2010 IMO Shortlist C4

Six stacks $S_{1}, \ldots, S_{6}$ of coins are standing in a row. In the beginning every stack contains a single coin. There are two types of allowed moves: Move 1: If stack $S_{k}$ with $1 \leq k \leq 5$ contains at least one coin, you may remove one coin from $S_{k}$ and add two coins to $S_{k+1}$. Move 2: If stack $S_{k}$ with $1 \leq k \leq 4$ contains at least one coin, then you may remove one coin from $S_{k}$ and exchange stacks $S_{k+1}$ and $S_{k+2}$. Decide whether it is possible to achieve by a sequence of such moves that the first five stacks are empty, whereas the sixth stack $S_{6}$ contains exactly $2010^{2010^{2010}}$ coins. Same as Problem C4, but the constant $2010^{2010^{2010}}$ is replaced by $2010^{2010}$. (Netherlands) Answer. Yes (in both variants of the problem). There exists such a sequence of moves.

六堆 $S_{1}, \ldots, S_{6}$ 硬币排成一排。一开始，每叠硬币都包含一枚硬币。允许的移动有两种类型： 移动 1：如果 $S_{k}$ 与 $1 \leq k \leq 5$ 堆叠至少包含一枚硬币，则可以从 $S_{k}$ 中移除一枚硬币，并向 $S_{k+1}$ 添加两枚硬币。步骤2：如果 $1 \leq k \leq 4$ 的堆栈 $S_{k}$ 包含至少一枚硬币，那么您可以从 $S_{k}$ 中取出一枚硬币并交换堆栈 $S_{k+1}$ 和 $S_{k+2}$。决定是否可以通过一系列此类移动来实现前五个堆栈为空，而第六个堆栈 $S_{6}$ 恰好包含 $2010^{2010^{2010}}$ 硬币。与问题 C4 相同，但常数 $2010^{2010^{2010}}$ 被替换为 $2010^{2010}$。 (荷兰)回答。是的(在问题的两种变体中)。存在这样的一系列动作。