For the Math-Nerds

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)?

For the Math-Nerds

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)?

Oof. I feel like thatâs probably some pretty nasty math. I mean, the chance of any four given skyfall gems being the same color is pretty simple; thatâs just (1/7)^4, or 0.00042, or 0.042% (Iâm 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, itâs not even 8AM and I havenât gotten to coffee yet, but I canât even begin to guess at how to quantify that. Best I can do is âsomething higher than 0.042%â. ;p

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 donât really care to reproduce the entirety of the match detection and board spawning algorithm for this question.

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

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 donât get an extra turn, thereâs a 95% chance that the AI will get at least one 4 gem match from the board.

I canât 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%. Itâs not 1/(7^4) because the color of the first gem doesnât 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 youâd 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 wouldnât know where to begin modeling.

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

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Stupid question incoming!

Does the kingdom affect the probability of gem colors that drop / form? Isnât Whitehelm more likely to have yellow gems dropping than the others or am I completely out of radar?

Itâs 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.

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This could be answered with a really neat feature:

Itâd be cool to have a âplayer statisticsâ page that tracks little factoids like ânumber of gems explodedâ, âturns playedâ, ânumber of extra turnsâ, etc. I want it to see if my average number of turns is significantly different from the CPUâs, but it could track things like this.

I find it hard to characterize what counts as a âskyfall extra turnâ 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 Iâm 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 âskyfall involvedâ set of extra turns and we have our answer!

I canât see the devs ever implementing this kind of player-specific statistics page, but I can dream!

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.

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Wow. Thatâs a lot higher than I expected, but it also explains a lot.

WellâŚ as a human maybe 1 in 4 oddsâŚ the AI has about a 95% rate it seems.

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