# 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?

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.

• If there's more than 9 mines and they make a 3*3 square then, the one in the middle is very hard to find except if you already discovered the rest of the game and know there's a mine left – Kii May 21 '17 at 14:17
• What counts as "not guessing"? Are you allowed an initial guess? I think what you meant is that you are also given a starting square which you know is safe, and then you can solve the puzzle from that square alone? – MaudPieTheRocktorate May 22 '17 at 3:19

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. $$\begin{array}{c|ccc} & A & B & C \\ \hline 1 & ? & ? & * \\ 2 & 1 & 3 & * \\ 3 & 0 & 1 & 1 \end{array} or \begin{array}{c|ccc} & A & B & C \\ \hline 1 & ? & ? & 1 \\ 2 & ? & ? & 2 \\ 3 & 1 & 2 & * \end{array}$$