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I have stared at this board for way too long and can't figure out how to move on.

Minesweeper board

As far as I can tell, the main problem is the bottom part with the double stacked '4'. It seems to me that I have multiple possible solutions there that would all work out, so I can't decide for certainty which one is the correct one.

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  • 3
    Very similar to gaming.stackexchange.com/questions/288247/…, which might be essentially a dupe. I don't think you can logically deduce a valid move, sometimes (a lot of the time!) this happens in the game of Minesweeper.
    – Timmy Jim
    Aug 17 at 21:42
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    Do you remember how many mines had already been placed? Or what the grid-size was?
    – Criggie
    Aug 18 at 9:46
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    Possible duplicate: Best move when they are no more certain moves left. Aug 18 at 15:58
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    If you want a Minesweeper variant that always produces levels that are solvable without guessing, there's Simon Tatham's Mines, or the much more unforgiving Kaboom by Paweł Marczewski.
    – Kroltan
    Aug 18 at 21:44
  • 1
    Minesweeper is not like grid-deduction puzzles, in that you do not have all the information needed to correctly solve it at the outset, and maybe never do until you try it, and therefore it nearly always comes down to some lucky guesses at the end, especially on the harder difficulties. Even the best players will lose more often than they win, just due to being forced to guess sometimes. Aug 20 at 15:46
35

I've been looking at this for several minutes, and I can't figure out any safe move to do.

I feel the only solution left is to guess, and the less dangerous move should be here:

minesweeper

That 2 already has an adjacent mine, so those 5 tiles in the green L-reversed shape only hide a single mine. This means that you, more or less, have 80% chance to do a safe move and get an empty slot. Hopefully, with another hint to let you progress.

I'll probably avoid the left-most tile, since you should have 50-50 chance to hit the mine there.

PS: Still stuck?

Edit after OP comment:
Another possible move is to choose one of the white tiles on the right, and since we don't know how many mines were remaining, let's use the default numbers:

30×16 grid is the Expert one with 99 mines, this means 480 tiles for 99 mines;
Resulting in 20.625% of the tiles hide a mine.

Which is pretty much the same chance of choosing a tile from the area highlighted in the above image.

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    After spending more time than I care to admit attempting to answer this, I arrived at the same conclusion. I will add that the lower right corner of the L seems the most probable location to survive with due to the 1 at the top. Aug 18 at 1:49
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    And also, the lower left corner is probably slightly more likely to be a mine because of the 3 to the left of it.
    – Arthur
    Aug 18 at 8:58
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    Great answer - do we know how many mines there are un-accounted for? The odds of a blind guess in the middle might be even better than the 1-in-5 - screenshot doesn't show the whole board though.
    – Criggie
    Aug 18 at 9:46
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    This raises interesting questions of probability. From the point of view of the orange-boxed 2, the five squares in green are equally likely (20%) to have a mine. But four of those five outcomes mean that the leftmost one (immediately below the red circle) is safe—in which case the one below that has a mine. So by that measure, the two next to the 3 aren’t a 50-50 shot, but 20-80! Aug 18 at 13:34
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    Fantastic answers. This was the result of me thinking "1 quick game of minesweeper before sleep can't hurt". Needless to say, I wasted way too much time. Unfortunately I rage quit last night so I dunno how many mines were left, all I know is it was a 30x16 grid.
    – wizard003
    Aug 18 at 14:30
36

As others have said, you’re right: multiple solutions are possible, and so there is nothing to do here but guess. Pinckerman’s answer points out a range of five squares that, in theory, are equally likely to hold a mine, giving you an 80% chance of survival. Here’s some more analysis of those possibilities.

Colour-coded outcomes of a 1-in-5 Minesweeper guess

I’ve allocated each of the five squares a coloured dot—if that square is a mine, then so are the other squares marked with the same dot. As you can see, there are some squares that should definitely be avoided, as they hold mines in three or even four of the five scenarios! (Squares not marked with any dots are independent of these five possibilities.)

However, it’s only an assumption to say that each of the five scenarios is equally likely. After all, from the point of view of the “3” in the lower left of my image, the two squares to the right are equally likely (50%–50%) to hold a mine… but the above analysis suggests that it’s actually 20%–80%. It makes sense that other factors might likewise change our risk assessment of the five squares to no longer be equally likely.

One possible factor is the number of mines remaining to be found. The purple, green, and blue scenarios each flag four mines, while red and orange only flag three. Other areas in your screenshot (mostly cropped from mine) give us three to five other mines. Does this alter the probability of any given outcome? I dunno!

In the unlikely scenario that there are exactly six mines (the bare minimum) left to be flagged, it may in fact be solvable without guessing. But I doubt it.

…And I was right to doubt it. As NotThatGuy points out, there is no situation in which the two lowest squares can be anything but a fifty-fifty guess.

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    Good analysis and nicely explained, +1
    – pinckerman
    Aug 18 at 14:23
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    If there's only a small number of mines, pressing in the big blank area is likely to yield information faster.
    – Joshua
    Aug 18 at 16:08
  • I disagree with the probability. Just as you disagreed that its not 50-50, it's not 80-20 either, all you did was change the direction but used the same formula. The formula used should give the same answer going left to right as it does right to left. This delves into the world of conditional probabilities.
    – rtaft
    Aug 19 at 13:33
  • @rtaft That’s what I was trying to say (in the paragraph starting “However, it’s only an assumption…”). It’s (probably) neither 50-50 nor 80-20. I’m working on this more in my free time, as and when I have it! Aug 19 at 14:03
  • Yep, apparently it's better to try that 50-50 guess at first. It will likely end with the whole board solved but those 2 tiles...
    – pinckerman
    Aug 20 at 16:10
25

A lot of the existing answers consider strictly the probability of any given cell being a mine, but seem to be missing what happens after you make a guess. You may be forced to make additional guesses, which would decrease your chances of winning quite significantly.

Of course a lot of this will depend on what you actually find, but there are some cells that seem promising than others. Below is the best I can come up with.

I'm ignoring how many mines are left (which also wasn't provided), but this can also be an important consideration.

You should pick either of the bottom-most 2 cells.

This may not seem like a great choice as it may not be the least likely to be a mine and it's not that likely to open up the board.

But nothing else being revealed will give you any information whatsoever about which of those cells you should pick, so it's a guess you're going to need to make regardless. This is always the case when you have 2 neighbouring cells containing only 1 mine with 3 mines (or a wall) on both sides.

This is objectively the best guess you can make if your goal to win.

If you need to make another guess, I'll probably suggest this:

Start off opening the cell marked "1". This should have about a 1-in-3 chance of being a mine (depending on how you calculate it, of course).

This guarantees that "2" isn't a mine either, since it's the 3rd cell next to 2 1s.

Because of the cell to the left of "1", we know only one of the cells above and below "1" is a mine. This means "1" is 1-4. If it's 4, we know "3", "4" and "5" are mines. If it's 1, this is the best case, since "3", "4" and "5" are all safe. It being 2 or 3 won't really be useful.

If "3" isn't a mine, it can either be 1 or 2. If it's 2, "6" is a mine. If it's 1, "6" is safe.

If "6" isn't a mine, it can be 1 or 2, which will determine whether "7" is safe or a mine.

If "7" is safe, this combined with "2" will tell us whether 8 is a mine, which can lead to opening up the whole upper area.

Having things work out exactly like this is fairly unlikely, but it seems much more promising than any other choice, which involves going much more into the unknown. Having that wall right next to you is good and bad: it means there aren't any potential mines in that direction, so the adjacent cells have fewer possible values, but you're also limited in how much you can open up, since you can't expand in that way.

Unless you have a more compelling guess, I'd generally recommend making a guess somewhere in a straight line next to open cells (the longer the line the better). There can be some guaranteed chains (like "2" above) and 1 cell can often give information about multiple cells behind it (like "3", "4" and "5" above).


If I need to make another guess, I may go with the cell below "1", since if the cell above "1" is safe, the 3 cells in front of the 3 below would all need to be safe, which just doesn't seem particularly likely (which may also support the above guess).

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    And note that if you lose the bet on the spot you picked you don't waste any more time on the board. If you're going to lose anyway might as well be as soon as possible. When I used to play Minesweeper I would always take such guesses. Aug 19 at 3:44
  • I remember playing MineSweeper a lot when younger, and I generally adopted a similar strategy: if I have to take a guess, then I'd prefer a binary output (make the board deterministic or die) to just rolling into another guess. Worst case, I take my loss, and move on to a more interesting board. Spending all that time clearing most of the board just to end up having to take the 50%-50% guess on the lower 2 cells anyway is just waffling. Aug 19 at 15:19
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The rim consists of 4 isolated groups not affecting mine placements within other groups.

Of those, 3 are simple dipoles.

In the more complex group, all five second mine locations for the key "2" results in valid boards, with no overlap between them.

Since there are three groups with varying possible mine density (two 2-1 dipoles, one complex 3-4), the board is only deterministic if the number of remaining mines is at its minimum, in this case exactly six mines.

dipole groups

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This is a breakdown of the non-trivial possible mine patterns. A given pattern family has a shape, and each pattern has a colour.

enter image description here

There is no deterministic way to find the next spot.

But we can seek to maximize information at minimal risk.

By far the most likely place for there to be a mine is between that 2 and 4 near the bottom. Assume there is a mine there. Now click on the 2 green circles next to it.

If red, blue or either of the orange circles are the real mine pattern, we live. And we get information about the pair of orange circles under the 2, and can start advancing, and we get to clear more territory regardless of what the actual pattern is. We could get confused between red, blue and a rightmost orange, depending on where other unrevealed mines are.

If green was the actual solution, we die. Too bad, so sad.

It is possible that one of the other red/blue cells is less likely to have a mine than those 2 green ones. But the fact you can clear 2 cells and mark a mine and either mark another cell (solving everything) or clear another cell if you make this guess makes it good information to risk choice.

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