Question 28
Eight boxes numbered $1$ to $8$ are lined up next to an empty bag. Each box contains one token. Miha has a lot of extra tokens and he plays a game where the following two actions are allowed:
* removing $1$ token from box $i$ ($i \lt 8$) and adding $2$ tokens into box $(i + 1)$,
* removing $1$ token from box $i$ ($i \lt 8$) and moving $1$ token from box $(i + 1)$ into the bag.
The game ends when there are no possible moves left. At least how many tokens are in the bag at the end?
* removing $1$ token from box $i$ ($i \lt 8$) and adding $2$ tokens into box $(i + 1)$,
* removing $1$ token from box $i$ ($i \lt 8$) and moving $1$ token from box $(i + 1)$ into the bag.
The game ends when there are no possible moves left. At least how many tokens are in the bag at the end?