Monday, July 2, 2018

Minesweeper win rates: more data and open questions

I've blogged before about my mild (?) obsession with Minesweeper. It is, in my humble opinion, the Platonic ideal of a game--the perfect blend of logic, statistics, and luck. I was able to find a fantastic version for Android--called Minesweeper GO - classic mines game--that I highly recommend you download if you have any serious interest in the game. I won't bore you with all the details, but the stats and hints logic are second to none.

Screenshot of a portion of an expert
grid on the Minesweeper GO app.
I''m still not 100% of the ideal play;
what would you go with?
Anyway, with this new app, I've gotten some more data on my Minesweeper win percentages:
Beginner: 86.04% (after 953 games)
Intermediate: 79.83% (after 1145 games)
Expert: 38.76% (after 4422 games) [1]
Being me, one of the things I was interested in was that the beginner and intermediate percentages are quite close. And this despite the fact that the beginner board as 8*8=64 tiles versus the intermediate board's 16*16=256 tiles--a 4x difference. However, it turns out that they have an equal density of mines (10 mines on the beginner board, 40 on intermediate: both work out to 6.4 tiles/mine, or about 0.16 mines/tile if you want to think of it that way). So it makes sense that the win percentages on each would be pretty similar.

As for why they're different at all, I think there might be (at least) two answers: the main one is that with more mines, there's just more opportunity for them to clump together in corners where you have to guess at the answer. And then there's the fact that I've been playing with the feature (apparently first introduced in the Windows Vista version) of every first click being an opening (i.e., empty--equivalent to a zero) turned on in the app. That automatic opening will tend to be a larger percentage of the board in beginner than in intermediate, I think, and so you will have already cleared a greater proportion of the field from the get-go on the beginner level, giving you more of a head start.

All of that made me curious: what size board would lead to the highest possible win percentage? Obviously there are an infinite number of board sizes and mine densities, so to be more specific maybe I could simplify the problem and ask what square board size with a density of 6.4 tiles/mine (or as close to that as possible given the board size) could be solved at the highest rate? Maybe the beginner board is the ideal in that regard, but maybe a slightly bigger or smaller board could beat it.

And finally, I think I'm going to reset my stats in the app (*gasp!*) and see how these stats change without the automatic-opening feature turned on. I'm sure I'll post the certain-to-be-fascinating results on here ere long! :)


[1] Note that in my previous post, I reported a 33.9% win percentage on expert, and now it's 38.76%--what happened? I may have just gotten better at Minesweeper, or maybe I tend to make fewer mistakes tapping on a phone screen than I did clicking with a mouse. And that guaranteed-opening-on-the-first-click feature being turned on likely plays a part. But actually I'm surprised that the difference isn't larger: as I said in a comment on that post, I actually started keeping track of my monthly stats for a few years on a publicly accessible google spreadsheet. There, you'll note that even without the opening-on-first-click feature, I steadily increased my monthly winning percentage (even getting as high as 44.6% in June 2012!), with my average winning percentage over the last 10 months I kept track working out to 37.07%. So it looks like the auto-opening feature only amounts to a 1 or 2% difference on expert. I guess that makes sense since the grid is so huge (16x30) that a small opening every time at the beginning doesn't change much.

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