What is the chance of getting another move after clearing the board?


#1

For the Math-Nerds :wink:

Now that the exploders are really en vogue:

What is the Chance of getting another move after clearing the complete board with an exploder (like Worldbreaker or Gorg or whatever)?


#2

Pinging @lyya who I believe can do these calculations in her head


#3

Oof. I feel like that’s probably some pretty nasty math. I mean, the chance of any four given skyfall gems being the same color is pretty simple; that’s just (1/7)^4, or 0.00042, or 0.042% (I’m pretty positive that the devs have said drop chances after the initial board setup are equally distributed, barring combo-breaker shenanigans).

But in terms of applying that to any four given consecutive orbs anywhere on a full board skyfall? I mean, it’s not even 8AM and I haven’t gotten to coffee yet, but I can’t even begin to guess at how to quantify that. Best I can do is “something higher than 0.042%”. ;p


#4

Rather than calculating this analytically, I suspect it will be faster to do this numerically. Set up a Monte Carlo simulation to generate a million boards using the known gem probabilities (14% for the six colors and 16% for skulls, IIRC). Then determine how many of those contain a 4- or 5-match on them.

This will be a lower limit, since cascades off 3-matches can lead to extra turns. But it should be close, at least. And I don’t really care to reproduce the entirety of the match detection and board spawning algorithm for this question.


#5

And then redo it with the numbers altered by there being a storm in play.


#6

If Famine is filled for the AI. You have a 0% chance of getting an extra turn.
They do tend to happen more often when there is a color storm happening.
If you don’t get an extra turn, there’s a 95% chance that the AI will get at least one 4 gem match from the board.


#8

I can’t solve this problem analytically.

The chance of any 4 gems being of the same color are 1/(7^3), or 1 in 343, or about 0.29%. It’s not 1/(7^4) because the color of the first gem doesn’t matter.

Assuming that the outcome of results is independent then: there are 40 horizontal chances and 40 vertical chances at success (4-match), each with a probability of 1/(7^3). There are also L, T, and + shaped matches that would count as 5-matches. Each of the 36 3x3 block contains 9 such shapes, for a total of 324 chances of nonlinear 5-matches, at 1/(7^4) probability each. To fail them all would be (1 - 1/(7^3))^80) * (1 - 1/(7^4))^324), which makes the chance of getting an extra turn:

1 - ((1 - 1/(7^3))^80) * (1 - 1/(7^4))^324)) ~= 0.3083 ~= 30.83%.

However, the premise is flawed, as these chances are not independent. If in any given row, gems ABCD are not a 4-match, then the odds of BCDE being a 4-match are considerably lower, as the only chance they have at matching would require BCD to share a color. Thus you’d think the chance is considerably lower than 30.83%. However however, none of this accounts for 4-matches from cascades, which are so complex that I wouldn’t know where to begin modeling.

Ultimately, @Grundulum is correct, and a simulation is the best predictor of such a complex system.


#9

Stupid question incoming!

Does the kingdom affect the probability of gem colors that drop / form? Isn’t Whitehelm more likely to have yellow gems dropping than the others or am I completely out of radar?


#10

It’s worse than that. Any L, T, or + shape has a significant chance to be part of a linear 4- or 5-match, since you already know that three consecutive gems are the correct color. This is absolutely solvable analytically, but there would be sooo many cases of overlap and double/triple/quadruple counting.

Edit: @Bolatsi. As far as we know, kingdom and banner have no impact on drop rates.


#11

This could be answered with a really neat feature:

It’d be cool to have a “player statistics” page that tracks little factoids like “number of gems exploded”, “turns played”, “number of extra turns”, etc. I want it to see if my average number of turns is significantly different from the CPU’s, but it could track things like this.

I find it hard to characterize what counts as a “skyfall extra turn” though. I can make a good sounding definition like:

A greater-than-4 match occurring as at least the 2nd match of a turn where at least one of the gems involved was not on the screen at the time the match occurred.

But I’m imagining a scenario where you trigger a 3-match that causes a skyfall 3-match that causes some gems that were already on the screen to make a 4-match. Hmm. Maybe the better definition is algorithmic:

1. All gems start with a "from skyfall" flag set.
2. At the start of each turn, unset the flag for every gem.
3. When a match is made:
    3.1 For each cascade past the first:
        3.1.2 If a gem involved in this match has the "from skyfall" flag set,
                set a "skyfall involved" flag.
4. At the end of the turn:
    4.1 If "skyfall involved" and "extra turn" are set, this was a skyfall extra turn.

Compare your total number of non-extra turns to this “skyfall involved” set of extra turns and we have our answer!

I can’t see the devs ever implementing this kind of player-specific statistics page, but I can dream!


#12

After running 10 million trials with randomly generated boards, the Monte Carlo odds of getting a native (non-cascading) Extra Turn appear to be approximately 24.8%. This is assuming even distribution of all gem types, including skulls.


#13

Wow. That’s a lot higher than I expected, but it also explains a lot.


#14

Well… as a human maybe 1 in 4 odds… the AI has about a 95% rate it seems.