Question 128
Each of the six boxes $B_1, B_2, B_3, B_4, B_5, B_6$ initially contains one coin. The following two types of operations are allowed.
Type 1: Choose a non-empty box $B_j$, $1 \le j \le 5$, remove one coin from $B_j$ and add two coins to $B_{j+1}$.
Type 2: Choose a non-empty box $B_k$, $1 \le k \le 4$, remove one coin from $B_k$ and swap the contents of the boxes $B_{k+1}$ and $B_{k+2}$.
Is there a finite sequence of operations that leads to a state in which the boxes $B_1, B_2, B_3, B_4, B_5$ are empty and the box $B_6$ contains exactly $2010^{2010^{2010}}$ coins? (By definition, $a^{b^c} = a^{(b^c)}$.)
Type 1: Choose a non-empty box $B_j$, $1 \le j \le 5$, remove one coin from $B_j$ and add two coins to $B_{j+1}$.
Type 2: Choose a non-empty box $B_k$, $1 \le k \le 4$, remove one coin from $B_k$ and swap the contents of the boxes $B_{k+1}$ and $B_{k+2}$.
Is there a finite sequence of operations that leads to a state in which the boxes $B_1, B_2, B_3, B_4, B_5$ are empty and the box $B_6$ contains exactly $2010^{2010^{2010}}$ coins? (By definition, $a^{b^c} = a^{(b^c)}$.)