Esoteric Discussion of Mine Game Analysis and Theory

[JackSonova]

Originally inspired by this post which got me to thinking about my own "double-up" strategy for Mines, and just how many ways you can play Mine in different ways as part of a double-up strategy, I began to wonder what lead people's motivations for their personal strategies because, most of us who have played Mines for some time, would likely feel to a degree that their own strategy is ideal or most suitable for their needs, and so when reading that other post, it surprised me to read someone's strategy that completely went against one of the major principles of why my strategy for doubling up was so different from theirs and was somewhat inspiring. So, after writing all that you will find below because of that inspiration and curiosity, I realized that it went way off tangent from what the OP originally made their post about (maybe, probably) and thought I'd move it to it's own thread to invite discussion from other players who might be very analytical, meticulous, or just simply thinks they're a Mines pro and would be willing to share their perspective on it.

A sort of thread and place of discussion of other player's logic and breakdown of their own personal strategies or perspective and approach on playing Mines.

 

One of the Intriguing Thing About Mines

I'd say for any "double-up" strategy, mines has a vast selection of  combinations to achieve this and in however someone preferences to set the number of mines and squares they aim for each round, it all boils down to tradeoffs of risk and reward. But, how accurately are we factoring in that risk?

Although, depending on how you look at it, any of the available choices of combinations available that offer a winning multiplier starting with at least two times your bet amount, all more or less merely give slightly better or worse odds statistically speaking. And, in a sort of over-analytical, semi-psychological, and part philosophical way, it would be interesting to wonder how we might distinguish the difference in people's risk tolerance and how we might have different perceived values when internally measuring our preferred taking on of those risks, especially in comparison to the statistical probability. Perhaps in some way, it's reflective of what we believe we have more control over or stand a better chance in if having to play the odds.

That is to say, for every game setting configuration whose outcome pays between 2x to 2.99x, the statistical probability of success for each variation can vary among the different multipliers and I've never stopped to think about how different they actually are as it might reveal one's affinity to how one might mitigate what feels is the best/easiest/or lucrative to play and gives us maybe an internal feeling of having a better chance even if mathematically, it may or may not.

The 2.xx's Multipliers, So Many Choices

Take for example all the possible outcomes that result in a multiplier of anywhere between 2.0x to 2.99x and notice the differences between the number of turns needed to achieve a multiplier for any of the 2.**x increments:

*(Please note that these multipliers may not be up-to-date and/or may be inaccurate from the time of this posting as BC.Game may change the payout at any time without notice, some values were rounded to the nearest second decimal place)

[Mines Setting] : {[Number of gems revealed] - [Multiplier]}, {repeat for other 2.0x-2.99x multis}

• 1 Mine : 13 - 2.06x, 14 - 2.25x, 15 - 2.48x, 16 - 2.75x
• 2 Mines : 8 - 2.18x, 9 - 2.47x
• 3 Mines : 5 - 2.0x, 6 - 2.35x, 7 - 2.79x
• 4 Mines : 4 - 2.09x, 5 - 2.58x
• 5 Mines : 3 - 2.0x, 4 - 2.58x
• 6 Mines : 3 - 2.35x
• 7 Mines : 3 - 2.79x
• 8 Mines : 2 - 2.18x
• 9 Mines : 2 - 2.47x
• 10 Mines : 2 - 2.83x
• 13 Mines : 1 - 2.06x
• 14 Mines : 1 - 2.25x
• 15 Mines : 1 - 2.48x
• 16 Mines : 1 - 2.75x

If we take 2.47x and 2.48x, this multiplier can be achieved with the following game modes and outcome, shown with their calculated probability of success: 

• 2.48x (exact is 2.475x) finding 1 gem among 15 mines; 10 / 25 = 0.4 or 40% chance of success
• 2.47x (exact is 2.475x) finding 2 gems among 9 mines; (16 / 26) x (15 / 24) = 0.64 x 0.625 = 0.4 or 40% chance of success
• 2.47x (exact is 2.4749x) finding 9 gems among 2 mines; (23 / 25) x (22 / 24) x (21 / 23) x (20 / 22) x (19 / 21) x (18 / 20) x (17 / 19) x (16 / 18) x (15 / 17) = 0.4 or 40% chance of success
• 2.48x (exact is 2.4752x) finding 15 gems among 1 mine; (24 / 25) x (23 / 24) x (22 / 23) x (21 / 22) x (20 / 21) x (19 / 20) x (18 / 19) x (17 / 18) x (16 / 17) x (15 / 16) x (14 / 15) x (13 / 14) x (12 / 13) x (11 / 12) x (10 / 11) = 0.4 or 40% chance of success

 

We can see that each of the 4 games mentioned have just slightly different variations in their multipliers but have  exactly the same probabilities of success. So, if you were someone who was aiming to get around 2.47x any option from the four above, you would have the same statistical probability of success while likely having a preference of playing one game setting over another and, to be exact, finding 15 gems when there is 1 mine is just slightly higher than the rest (a whopping 0.0002-0.0003x higher).

Comparing 2 Ends of the Spectrum

Now if we compare the two ends of the spectrum for a more distinct contrast, then we'd have the following:

• 2.00x (exact is 1.9976x and 1.9975x respectively) for finding either 5 gems among 3 mines or 3 gems among 5 mines; 
  • (22 / 25) x ( 21 / 24) x (20 / 23) x (19 / 22) x (18 / 21) = 0.495652173... or 49.57% chance of success
  • (20 / 25) x (19 / 24) x (18 / 23) = 0.495652173... or 49.57% chance of success
• 2.79x (exact is 2.7909x and 2.7905x respectively) for finding either 7 gems among 3 mines or 3 gems among 7 mines;
  • (22 / 25) x (21 / 24) x (20 / 23) x (19 / 22) x (18 / 21) x (17 / 20) x (16 / 19) = 0.354782608695652173... or 35.48% chance of success
  • (18 / 25) x (17 / 24) x (16 / 23) = 0.354782608695652173... or 35.48% chance of success

Interestingly you might notice that for each inversely paired game setting, the multiplier is insignificantly higher in the game with a lower amount of mines even though both games offer the same exact probability of success. But more interesting is why maybe some would rather do 5 gems at 3 mines (like the one mentioned in this post) as opposed to 3 gems at 5 mines (such as myself--if playing for 2.00x).

For me, although for a double-up strategy, I don't play 5 mines too often, in my personal opinion there's more opportunities to fail when electing to set the game to 3 mines and having to choose 5 squares successfully to achieve a 2.00x multiplier, even though each square selection has a lower probability of failure per se. In a way, you could say that I'd prefer "being luckier" in fewer turns per game even though mathematically it works out to be the same exact odds. This of course, assumes that we're not going to consider a 0.0001x higher multiplier a big deal. It's kind of weird if you stop and really think about it, as in, what we're more comfortable with or prefer, how we perceive our own ability to win such that maybe the we feel more capable or luckier by selecting 5 gems among 3 mines while some may feel more comfortable selecting 3 gems among 5 mines. So while we know statistically speaking they measure out to be the same, what motivates us to prefer one over another? Is it our own delusion to think we might be more capable of selecting 3 squares correctly rather than 5? Or, what sort of logical structure in our thinking leads us to even have a preference? Kind of going off tangent here but just some food for thought I guess.

The Standards We Set & What We Prefer

When deciding between 2.00x or 2.79x in the above example, we’d be looking at a difference in probability of success by 15 less percentage points while gaining a 39.5% higher multiplier applied to our potential winnings. 

While clearly this is simply a trade-off between risk and reward, what adds some intrigue to ponder is if you took the 2 games available that require the same amount of gems be found to succeed. Then, you would have to decide on playing a game requiring finding 3 gems among 5 mines for 2.00x or finding 3 gems but among 7 mines for 2.79x, what would you do or what do you take into consideration?

More risk, more reward, of course, but how or what we factor into our decision making may reveal some elements we take into consideration when perceiving probability and risk that is perhaps not subject to our own logic, but our own understanding and interpretation of the world. Such that we may be considering how:

• Both require selecting only 3 squares successfully
• 0.79x higher nearly makes it a triple-up
• It’s only 2 more mines or possibilities of failure being added, for a total gem to mine ratio of 4:5 ( a board of 80% gems) versus 18:25 (a board of 72% gems)
• Less mines on the boards means less chance of choosing the wrong square 
• etc.

These add subjective and personal values in the risk assessment of each game mode that can’t always be necessarily justified or communicated to someone else.

Only the Beginning

Needless to say, these particular 2.xx multiplier game strategies are just one of the many available style game plays Mines can be enjoyed as and when we broaden the topic to something more general such as, “How do you like to play mines?” we’ll be opening the door to completely different approaches and perspectives on the game that will alter how we might see the game as it is now.

With it’s auto-play feature and insanely high potential multipliers, I bet there are lots of considerations I've yet to think of to take into account: for instance, those who play for the challenge and recognition of achievement, where risk isn't so much a factor as much as guessing a prediction somehow and what sort of ideas stem from that. 

With Mines, there’s likely to be plenty of unspoken self-quantified theories or understandings of how the game is best played and if you find the conversation intriguing, feel free to jump in and share how you like to play the game of Mines!