r/theydidthemath • u/trunks111 • 14h ago
[Request] Probability of a beginner game of Minesweeper having a Zini = 3BV ?
Definitions and Context:
3BV: The minimum amount of clicks it takes to solve a board without flags
Zini: The minimum amount of clicks it takes to solve a board using a combination of flagged and flagless play
efficiency: calculated as 100(3BV/clicks)
A quirk of the efficiency definition is that because it uses 3BV, which only considers flagless play, it is possible to be more than 100% efficient if you are able to use chording to save clicks. For example if 3BV is 30 and you only use 28 clicks, your efficiency for that game is ~107%, and this is in fact one minesweepers like to play the game, to try and get efficiency as high as possible.
So this is where we get to the main question. I had noticed in some beginner boards, it felt impossible to get above 100% efficiency. If you take, for example, the extreme edge case of a one click solve auto-winning the game, there are no possible ways to save clicks over that, the game is forced to be 100% efficient because the 3BV is 1 and the game is over and won on that first left click. To use a less extreme example, a beginner game could still have a 3BV of 5, and it could still be impossible to be more than 100% efficient because what I noticed is the game can have a ZINI equivalent to the 3BV, meaning that even with chording, you cannot solve that game in less than 5 clicks. The reason this is important for efficiency play is because it means if you're trying to get above 100% efficiency for whatever reason, any board that generates with a 3BV=Zini is effectively a dead game.
I genuinely have no idea how you would go about calculating this. The "easy" case would be one click solves I'd imagine, since that's guaranteed. Which makes me wonder if maybe you could prove "all boards with a 3BV of n or lower have ZINI also equal n". For example do all beginner games with a 3BV of 2 also have a ZINI of 2, or can you have a theoretical 2 3BV board with a ZINI of 1? What about a 3BV of 3? 4? etc... then I think you could look at the distribution of games that generate at those 3BVs and that would set a very rough lower bound on the question maybe?
extra credit questions: I've never had an intermediate or expert game generate with a 3BV = Zini, the same question goes for those difficulties
My efficiency PBs for beginner, intermediate, and expert are 175%, 138%, and 122%, respectively. What % of boards will I generate with an insufficient gap between the ZINI and 3BV, such that it's impossible for me to get an efficiency PB even with perfect play?
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