Generate a minesweeper board which doesn't need guessing

Question

I am designing a Minesweeper-like game (with modified rules), and I want to prevent player from guessing. My goal is: The generated board is with few revealed squares, and p

Answer 1

Disclaimer: This may or may not be entirely correlated, but it's something neat I noticed, and might be useful to others trying to find the answer.

In minesweeper, there's a neat little thing I found when looking at different minesweeper boards: In the game, when generating a board, all non-mine squares have their value set to the number of neighboring mines. However, you can apply the same thing, but in reverse: All mines add 1 to the value of each neighboring space. This makes it possible to think of mines less like mines, and more like layered 3x3 squares. To demonstrate, here's a board, stripped of any neighboring mine counts:

....x
.x...
x..x.
..x..
x...x

If we use the normal generation method, setting the value of each non-mine tile to the number of neighboring mine tiles, we get:

1111x
2x222
x33x1
23x32
x212x

So what's the issue with using this method? Well, it has to do with just how many times we're checking for a mine. Let's see:

1111x  -  took 23 checks
2x222  -  took 31 checks
x33x1  -  took 26 checks
23x32  -  took 31 checks
x212x  -  took 20 checks

Ouch. That comes in at a total of 131 checks. Now, let's try the square method:

1111x  -  took 8 checks
2x222  -  took 13 checks
x33x1  -  took 18 checks
23x32  -  took 13 checks
x212x  -  took 11 checks

This method is much faster, with a total of only 63 checks, less than half of the naïve method.

These "squares" can also be utilized when solving boards. For example:

0000
?110
??10
???0

In this example, we can clearly see a corner, and therefore a square, with a mine in the center (+ and - are queued opens and flags):

0000
?110
?-10
???0

We can also expand the corner, since there isn't another square on top of it:

0000
+110
+-10
?++0

However, we cannot expand the last ? in the same way until we uncover all the queued tiles. Just as an example, I'm going to use some twos:

0000
1110
2x10
?210

Now we can see another corner, and it turns out that it intersects a mine. This can be hard to spot, but this is a corner.

One thing to watch out for, though, is something like this:

00000
01110
12x10
x?210
2x100

In this scenario, there are 3 squares. The 4x4 region in the top right matches the previous scenario, but don't get fooled: the twos from earlier are part of two separate squares in this one, and the unknown tile happens to be a three. If we had placed a mine there, the twos would be complete, and we would have lost to misjudgment, trying to uncover a tile that seems safe.

TL;DR: Mines can be thought of as layered 3x3 squares, which can save time when generating and/or solving.

Answer 2

It's well-known solving minesweeper is NP-complete.

This is true but perhaps not as relevant as you think. The proposed algorithm is something like "repeatedly generate random boards until the computer can solve one". NP-hardness is a property of the worst case, but here we're really interested in the average-case hardness. If an unusually hard board is generated, we can time out the solver and restart with a new board.

Also, even if there were an oracle to distinguish good boards from bad, would you really want the user to have to solve a hard problem in order to avoid guessing? A less talented computer solver might bias the choice toward fairer boards.

Answer 3

The implementation of Minesweeper in Simon Tatham's Portable Puzzle Collection is guessing-free. (It's also MIT licensed, so you're free to copy his implementation if you so desire.)