# Is there a definite answer to this Minesweeper puzzle? If not, which square has the best chance of being safe?

I was working on solving an Expert puzzle in Minesweeper, but came across a hurdle in this corner (the top left).

Is there a definitive solution to the problem? If so, what is it? If not, which square should I tap to increase my chances of not tapping a mine?

There are 5 or 6 mines in this segment. (There are 6 mines overall, but one of those mines could be in a different corner, not in this screenshot, that's totally surrounded by mines, so I'm unable to tell whether or not that square is a mine.)

## 2 Answers

hexomino is correct that there's not a safe way to proceed, but the heuristic doesn't give the right probabilities for the remaining squares.

Looking at the top five unknown squares, there are five possibilities for that region, each of which corresponds to exactly four solutions for the whole grid. These are (x=mine, o=empty):

oo   xo   ox   oo   xo
xx   xx   xo   ox   ox
o    o    x    x    x


This gives probabilities (left to right, top to bottom) of 2/5, 1/5, 3/5, 4/5, 3/5.

However, that's not really relevant.

The four squares in the middle contain two squares along a diagonal, and the two squares to the bottom of the group of seven contain exactly one mine. There will never be a way to figure out these six squares with more than 50% chance even if you solve the rest of the mine.

However, guessing one of those groups can solve both (if you guess the right way), and also tell you about the bottom square in the group of five. So the right thing to do now (in terms of maximising your probability of clearing the grid) is guess that the two squares at the bottom of the group of seven have a mine on the left-hand side and an empty square on the right. If this works, you have got information about the four squares diagonally below and the one immediately above. Since you can't avoid this 50% guess, you haven't lost anything if it doesn't work.

This strategy gives a 40% chance of winning. If you try the "safest" square first, even if it's not a mine you still won't be able to distinguish between the first and last configurations above, and you'll still have to guess in the other six squares, meaning you can't do better than 30%.

The square marked with an 'X' in the image below is definitely a mine.

Reasoning

If it is not, then both squares below the '4' must be mines but that won't be a valid solution since it implies the two squares below those are not mines, which is not true.

Next move

Click on the square above the 'X'

After the hint

There is no unique solution but among the three squares highlighted in red, two must contain mines

This would suggest that each of these squares has a $\frac{2}{3}$ probability of containing a mine.
As a result, the square to the left of the '2', just above this configuration has probability $\frac{1}{3}$ of containing a mine. Also, since you don't know if there are $5$ or $6$ mines remaining, the top left square also contains a mine with probability $\frac{1}{3}$ so you should go for one of these two (each other square contains a mine with probability $\frac{1}{2}$).

• You early bird :) I was just updating the image to solve this portion of the problem. What's next? – gparyani Sep 10 '18 at 10:16
• I've updated the answer. The solution now is not unique but the top two squares have lower probability of containing a mine than the rest. – hexomino Sep 10 '18 at 10:31