# Color Stacking: The Odds

JK thanks for the math. That helps a lot

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Weird. As someone else linked, I tracked the initial boards only, and I didn’t find anything suspicious. Did you count the tiles that come from the bottom, too? Manual or programmatic counting?

Thank you for this info, at least something actually shows that the board do miss out the color wanted.

But where are these probabilities coming from, from set of 1000 boards? Or from the devs? How credible is the data? Just to be clarified.

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You’re very welcome. The numbers I’m giving are pure math, using statistics. These are the expected probabilities of finding a given number of tiles of a certain color, for a random board of 35 tiles with 5 colors.

Click here for how to do the math, for those who are interested.

The color we’re stacking into has a probability of appearance of 20%, since there are 5 colors. The other 4 colors together have an 80% probability of appearance.

So we can calculate the probability of seeing X tiles of our stacked into color like this:

Probability = 0.2^X * 0.8^(35-X) * (number of ways we can pick X tiles out of 35)

That last part is called an N-choose-X problem. The number is given by:

number of ways we can pick X tiles out of 35 =
35!/((35-X)! * X!)

Here, the exclamation mark means the factorial operation. That’s where you multiply together all the numbers from 1 to the number next to the exclamation mark. So 4! = 4 * 3 * 2 * 1

The simplification here is that 35!/(35-X)! can actually be cancelled down to:

35 * 34 * 33 * … * (35-X+1)

So if X = 3, for example:

35!/((35-X)! * X! = 35 * 34 / (3 * 2 * 1) = 198.333

Putting all this together gives:

Probability = 0.2^X * 0.8^(35-X) * 35!/ ((35-X)! * X!)

So, let’s say we want to tell what the probability of getting only 3 tiles of our color is. We plug in 3 for X, and get:

Probability = 0.2^3 * 0.8^32 * 35 * 34 * 33 / (3 * 2 * 1)

So Probability = 4.15%

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Very interesting thanks.
But on a side note, i will bump this thread since its kinda related and i dont want to open another one just for this, i’ve read somewhere on a board, i dont think it was here on the forum, that when you’re staking like, 5 reds for example, you get a small bonus to the overall attack too, like basically if your attack power is 100% when having a monochrome team it will be like 105/110%. Is it true or just a myth?

*obviously i aint talking about abilities bumping the attack stat, just the pure total attack of a monochrome team.

You’re welcome! Glad you found it interesting.

Put in this way, it is just a myth. But there really is a big advantage to color stacking on tile damage, and it comes from the way the damage equation works (sorry if you already know this; but it may be new info to some people).

• Average Tile Damage = 33*(Attack/Defense)^1.35

When you stack colors, the attack stats of all of the attackers of the same color get added together to come up with that “Attack” number. The exponential in that equation is huge in it’s effect when you significantly increase the damage.

Let’s say you stack 3 of the same hero as your color stack. Then the tile damage for that color would be:

Average Tile Damage = 33*(3*Single_Attack/Defense)^1.35

We can actually factor the 3x out if the equation like so:

Average Tile Damage = 33*(Single_Attack/Defense)^1.35 * 3^1.35

Now, 3^1.35 = 4.4

So instead of your 3-stacked tile attack being 3x as powerful as a single-hero tile attack, it’s 4.4x as powerful. That’s huge.

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And turn the game into a garbage, no matter what you have, what you know or what you do. A team of 4100, with one of the best defenses in the game, who has just won 4 out of 5 in defense with their peers, sees as 2 SHITS, with capital letters, 4 * and with a better hero danceself I have just removed 120 cups . That makes the game, because of the stacking, in an absurd and meaningless lottery. SG PENALIZES STACKING ALREADY.

Stacking is overwhelming the game strategy, I agree. I’ll come back to that…

I honestly had no idea it worked this way. It explains so much.

It also reveals an easy solution;

Calculate each hero’s contribution separately and then add. Instead of doing the sum of attacks and then computing damage, compute damage, then sum. That moves the summation outside of the exponential calculation and greatly reduces the benefit of stacking.

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Great idea. It would also be a good idea for Sg to rescue a proposal made by a user about compartmentalizing the game and making divisions. Actually, a team of 3200, in mono and with a good board beats anyone, even number 1, but adulterates the game, because it can not defend beyond 2,300 or 2,400 cups, which is its level.

One major problem that I have with this proposal is that it has the unintended consequence of nerfing the damage done to titans. Not exactly something that should be taken lightly consider the possibility of stronger titans returning.

Furthermore, the way the current formula works gives player with shallower benches (mostly 4*, some 5*) an opportunity to defeat very good defenses (players with mostly 5*). I think you would see far more cup droppers to avoid having to deal with Guin/Grave. Rainbow doesn’t really help you if you are outgunned. A color stacked 4* team can beat a full 5* team. A rainbow 4* team will get thrashed by a full 5* team.

I’m not discounting that rainbow offenses with good synergy. All I’m saying is that the current color stacking formula provides an increase in the number of offenses that can beat a given defense. Changing the formula is a boost to defensive teams by eliminating some of those possibilities. If SG thinks that such a boost is necessary to fit their desired model, then I’d agree this is the way to go. Personally, I like the way the system is now, but would not be upset to see it change. I would simply ask them to also adjust titan defenses to account for this change.

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The titan, of course, should not change, but about the other is that’s what it is about. Do you think it’s normal for a 4 * team to beat the Top? And what is the effort or talent worth if you have a bug that allows everything. Why grow? Another solution that would help a lot in this, given how unpopular it would be to remove it and would also favor the new ones would be to change the raids, to put only 100 above or below, or 250 maximum of level, would equal the challenges.

I don’t see a problem with a 4* team beating the top if they make the gamble to color stack a 3/2 or mono team. It’s not that much different than using Wu Kong’s Gambler’s Stance in my eyes. Sometimes they miss, other times it is a rousing success.

Why grow? Because instead of beating them 1 out of every three times, eventually you’ll be strong enough to beat them 2-3 times out of every three attempts. Improving your roster will improve your success rate on offense and defense in raids. I think it is misleading to call it a bug. It is simply the mathematical result of the damage formula the game uses.

Thinking about it, it affects all aspects of the game, not just raids. Pushing for completion of rare quests, challenge events, titans. I simply worry about making a game-wide change because the top 1000 don’t like losing on raids to players they feel are inferior.

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So, I wasn’t expecting this degree of surprise at my statement. I thought this was the general understanding. Since it isn’t, I’m currently collecting data to verify that my understanding of the combination rule is how it actually works. Let me make sure I’m not getting it wrong before we worry too much about the in-game implications.

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4 against sniping or mana busting tanks and the 5th opposite to the own stack works nice.

2 against the tank and 3 against the hardest hitter, if it’s a weaker tank.

If the tank buffs hard, take a rainbow.

I’ve looked at this question a bit now, and I still don’t have a satisfactory answer. It’s clear from @SolemnWolf’s examples section that the attack stats used to be added together and then the exponential applied, exactly as I described above:

But in running my own tests on PvE, this doesn’t seem to be holding up. I can’t tell yet if the problem is:

1. They split out the attacks individually at some point, like Kerridoc suggested

or

1. There really is some linearization of the X^1.35 calculation as damage increases.

It looks, after about 80 sample points, as though #2 is more likely than #1. More research is definitely needed.

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I thought I’d throw in a few color-stack screenshots I took recently.

Here’s a blue 5-stack I used as a desperation move trying to knock out the tank in a recent war. The team was significantly stronger than mine, but because of the (very!) good luck on the starting board, I was able to not only kill the tank, but wipe the whole team. (Things like this are why I really enjoy this game!)

Here’s a red 5-stack I used, same as before, just looking to punch a hole in the middle for my allies to work down the rest of the team. I didn’t fare as well here, but Skittleskull was hurting after the second move and I think I knocked out 2-3 of this team.

And since there was some talk about stacking defensively, here is a team I faced. I built a stacked green team but then accidentally clicked to start with my usual rainbow team - I managed to win anyway.

Make of these what you will. I think the first clearly shows that the game isn’t designed to screw over people for color-stacking, as obviously I had a very good starting board on that one.

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Very nice! Thanks for sharing these.

I know you mean well and want to provide the best feedback to us all. The numbers look great, the stats and so on. My last 2 wars I used 6 color teams each, 75% of my teams never got to fight due to unbalanced boards. Yesterday I did 12 raids of which 10 had no color tiles/diamonds and ended with the same results.

Most of our Alliance says the same thing. It’s not about whining or claiming the game is unfair. It’s about loading a 5-0 team, and the tiles and diamonds don’t show. I change to a 4-1 team and a few more show up. I then move to a 3-2 team and it starts to balance a bit.

These are factual results, on our end of the game, as well as the games moderators stating their factual numbers. You have yours, we have ours.

I can take a loss, it’s just a game, but numbers being provided to us don’t always reflect the actual board results when playing.

No sarcastic replies needed like in the past. This issue has been closed in the forum, but still exists in the game whether you agree or not.

Thanks for all you do for the game, us and the experience of playing.

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I’m not really sure who you’re addressing your post to. If you mean me–and I’m the one who calculated those numbers–then you should know that I’m just a player like you. These numbers weren’t provided by Small Giant. They’re the correct numbers for a true random distribution of boards, though. I’d be happy to show you how to do the calculation, and you can do it yourself.

In your post, you keep mixing the words “tiles” and “diamonds”. It makes it somewhat hard to tell just what went on in those matches you’re talking about. I assume you’re not trying to say you literally got zero tiles of your strong color in 9 war matches.

If you’re saying instead that you had trouble making diamonds, though, that’s less surprising. It would also not be surprising that you had trouble filling your mana bar. That takes a lot of tiles to do when heroes aren’t Very Fast mana. A randomly distributed board is much less likely to have big clumps of a particular color. Mono stacking is a risky strategy precisely because of this. I wouldn’t recommend it.

Just to prove a point about what these boards tend to look like, though, I went and grabbed the first revenge-able enemy out of my watchtower and stacked mono against him. Here’s the opening board:

Then I grabbed the second revengeable-opponent. Here’s that opening board:

And then the third. Here’s that board:

I didn’t cheat or cherry pick. Those are pretty average boards.

I’d invite you to start taking screen-shots of your own boards and counting up tiles. If you do a large enough sample, I bet you’ll find that you have an average of 7 tiles of the color you’re stacking into. If you’re not counting tiles, then you don’t really know whether you’re averaging 7 tiles of your strong color or not.

Monocolor teams demand very accurate, careful tile play. You have to be much more careful, because there’s no second color to save you like there would be when you stack 3/2. It’s easy to get frustrated, or to get caught charging your opponent’s mana without making much progress on your own.

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color stacking is an option for high defense heroes or very fast mana only to me. my leader runs 3 -2 color stack pretty much every fight and being that he has a 4 k team he stays in high diamond with this strategy, but in wars he either wins big or loses big leaving a high team amost full health with none of our low guys able to clean that up which is not good in my eyes.

in my personal experience i only color stack RIGARD AS TANK, GRAVE AS TANK, RICHARD AND AZLAR AS TANK, AND KASHREK AS TANK. occasionally lianna and sartana need to be stacked against.

i understand color stacking benifits with tiles damage and i also understand how quickly your lineup can suffer immensly in wars because u have stacked yourself out of usable heroes round 1. this latter is why i try to teach players how to use their skills vs stacking as having 6 solid teams is better than one big kill.

my highest team is 3800 and we war with teams where 5 or 6 people are 39 plus and another 10 are around my level yet i finish top 5 of every war but 1 for the last 6 months.

i am an avid believer that constant color stacking reduces actual learned skill and can be more detrimental in learning counters and develop a heavy reliance on gem damage. me and Anchor from 7DD had a nice convo about this topic, but hey to each his own.

the only person i always stack against is gravmakers as tank, every one else i may go 2 2 1 or 2 111 but my main goal is to win with the most character avaiable in my roster to be lethal in both rounds of the war not just one.

you will also notice the more you stack in raids to beat teams 2 and 300 points higher which feels good intil every one of them owns you on revenge attacks.

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