Suppose we have an $m\times n$ minesweeper grid containing $k$ mines (for example the beginner grid is $8\times 8$ with $10$ mines). I have the following related questions:
- What is the probability of solving the grid if we play optimally?
- What is the probability of being able to solve the grid without guessing, if we are given that the first click reveals an opening?
- What is the expected number of mines that would be hit if we had infinite lives, if again we were to play optimally?
I don't hold out much hope for an exact formula for any of these, but any kind of approximation or bound would be good. How would we approach such problems?
I am aware of the option of using a customized minesweeper solver to play many games and get approximate answers via this method, and may well try this one day, but for now I am interested in the mathematical approach.
Perhaps a reasonable simplification would be to treat the grid as infinite with a mine density of $\rho$ (normally $\rho\approx 0.2$) and attempt to calculate the above problems for an area of cells (not necessarily in a rectangle?).
It's simple for very small grids or numbers of mines, or when $k\approx m\times n$, but I would like to get a rough idea for the standard difficulty levels which are: Beginner $8\times 8$ with $10$ mines; Intermediate $16\times 16$ with $40$ mines; Expert $16\times 30$ with $99$ mines.