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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.

Lê Đức Minh
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  • 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
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    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 03:19
  • By minesweeper rules, it is $nm - 1$, as it is solved after 1 click. – DanielV Nov 28 '23 at 16:42

3 Answers3

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

castor
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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.

Kii
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  • Yep, I also think that 7 mines is enough for any board bigger than 4 by 4, and that's enough because I don't really care about smaller board! – Lê Đức Minh May 21 '17 at 14:31
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The answer is actually... only $1$! Some puzzles with only $2$ mines require guess!!!

You know that the initial guess is not a mine. Let's say there are $2$ mines, and your initial click shows the number $1$. $1$ of adjacent spots have a mine, and $1$ of non-adjacent spots have a mine. You have to guess.

So the maximum number of mines that never needs guesses in any puzzle is $1$, if either $m$ or $n$ is $3$ or higher.

If $m=n=2$, The answer is actually $0$. In a $2 \times 2$ minesweeper, even only $1$ mine requires guess. If $m=2$ and $n=1$, there are only $2$ squares, so the answer is 1 again. With 1 mine, it means that whatever square you click for the first time, the other square is the mine, so no guessing and $100\%$ success.

Dominique
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Kgyo
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