# Minesweeper challenge #5

Coming back to another Minesweeper challenge...

I've run into another "unsolvable corner" (corner of a puzzle where there isn't enough information to be 100% certain that any particular square that isn't a mine).

Which squares in the below puzzle have the highest chance of being safe, and what are the chances I can fully solve the puzzle based on the clues present? There are 6 mines remaining.

Given your current situation, I believe that

You should fill in the two bottom middle boxes as being flags, because they are 100% mines. Then, the top two squares remaining contain at most 1 flag. The chance of guessing a safe square is 50% (see diagram). The three squares in column two that are touching the 4 each have a 1 in 3 chance of being a mine (there is definitely 1 mine somewhere in those three), so the chance of guessing a safe square is 67% (and the chance of being a mine is 33%. The remaining five squares have 2 mines, so there is a 40% chance of guessing the mine and a 60% chance of guessing a safe square.

I would therefore guess that

Take a guess at one of the three squares in column 2 that are touching the 4, that will give you the best odds.

Diagram:

• Ah, my eyes somehow skipped the two ones in the bottom right... – gparyani Mar 5 '19 at 1:13
• How did it go?? @gparyani – El-Guest Mar 5 '19 at 1:42
• @gparyani Doesn't the question also ask for the chance of fully solving the puzzle? – noedne Mar 8 '19 at 3:01
• @noedne That answer is provided here. – gparyani Mar 8 '19 at 3:02
• @gparyani What is the answer? I only see the probability of the first square clicked being a mine. – noedne Mar 8 '19 at 3:04

I'm surprised nobody other than noedne thought that the other answer wasn't complete. The question clearly wanted to know the odds of solving the minesweeper puzzle, a non-trivial question even at this size.

There are 60 states the mines can have in this puzzle. First thing to note is that the 50/50 pair at the top of the puzzle is always going to be a 50/50 shot. No clue from the opposite side will shed any light on it. Therefore it makes sense to guess there first because it is possible that we can learn something from it. I choose to click the left cell but it doesn't matter which you pick.

I'm not going over every decision I made, but there are a few places where guessing in the right spot might save you in the right scenario. For example, my second guess was commonly the middle-right cell. It has both better odds and better chances of being informative. I highlighted in yellow a cell that I "clicked" as a guess, but I know this diagram isn't as informative as I might like. If this question wasn't so old, I might have spent more time on the diagram.

Answer is: probability is 13/60 or 21.67% of winning, assuming you use my strategy. I'm fairly certain you can't do better, a formal proof would take too much time (I think).