For a m by n minesweeper board, what is the maximum number of mines can exist so that any puzzle with those mines is solveable without guessing?

Question

When I play the game Minesweeper, I make the puzzle more difficult by increasing the amount of mines and still keep the board. Once I set the maximum number of mines for a 9x9 board, which is $67$, I realise that the chance to win is almost zero(!). And when I play a game with a $24$x$30$ board with $150$ mines, I sometimes have to guess in order to win the game.

And after all, my question is:

Given a m by n Minesweeper board, what is the maximum number of mines can exist so that any puzzle with those mines is solveable without guessing?

Note: This question has some similarity to mine.

Answer 1

Two if the dimensions are large enough.

If you have three mines, may have to guess between B1,C1,C2 and A1,C1,C2. Or A1,B2,C3 and A2,B1,C3. $$ \begin{array}{c|ccc} & A & B & C \\ \hline 1 & ? & ? & * \\ 2 & 1 & 3 & * \\ 3 & 0 & 1 & 1 \end{array} or \begin{array}{c|ccc} & A & B & C \\ \hline 1 & ? & ? & 1 \\ 2 & ? & ? & 2 \\ 3 & 1 & 2 & * \end{array} $$

Answer 2

Considering a 3*3 square full of mines and another 3*3 square with 8 mines around a empty space. There's one chance out of two that you can guess which square has a mine inside the 8 mines. So I would say 17 mines.

This is a worse case scenario. It can be even worse considering corner positions. Maybe just 7 mines in two corners would suffice to make it a guess.