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brairtrainer

Dragcave Maths

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So I was in the irc one day and someone said they wanted to make a 100 gen even gen lineage, so I decided to find out the number of cbs you'd need, turns out that number is 1267650600228229401496703205376 individual cbs O.O. So I got to thinking there must be other interesting maths here.

 

Infact if you wanted to make a 100 gen even gen lineage you couldn't as there can only be 930909168 dragons over all on dragcave. Assuming 1/3 of all dragons are CB you'd need 18 digit codes. So here is what this thread is for, if you find something interesting and mathmatical we'll add it to trivia, if you know or want to know how many cbs you'd need for a lineage you can ask here and we'll update a list on the 2nd post.

 

Trivia:

 

Amount of CBs needed for a 100 gen even gen lineage: 1267650600228229401496703205376

 

The square root of a purple dragon is this color: user posted image

 

The length of time needed to do a 100 gen even gen lineage: 70877864144715090.941945943833268 times the age of the universe

 

Number of red dragons needed for never ending incubates on a gold trophy scroll: 49

 

Amount of posts a member could make per year if they posted every 30 seconds: 1051200 not accounting for lag

 

Approximate amount of help threads that go through help a year: 1652

 

If a user just started and was able to pick up eggs exactly when the old ones hatched (and he didn't abandon) would reach gold trophy: 279 days, and 502 dragons to show for it.

 

Amount of pinks needed to always be able to influence on a gold trophy scroll (not counting low timed eggs): 50 (25 of each gender)

 

It is possible to get to gold trophy in less than a year.

 

The square root of a neglected female dragon: user posted image

 

The square root of an ice dragon: user posted image

 

Number of reds required to remain hatchie locked on a gold scroll with an incuhatchuable AP: 105

 

 

Edited by brairtrainer

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"Could there possibly be that many CB dragons on DC already?  There are a lot of users on the site, and each of them has a bunch of dragons."

 

Short answer: No.

 

Long answer:

1267650600228229401496703205376... that's a lot of dragons! Problem is, there aren't very many possible codes for all those. And not every dragon with a code is CB. So how many characters long would the codes have to be for there to be 1267650600228229401496703205376 CB dragons?

 

Well codes can be uppercase letters (there are 26), lowercase letters (26), and digits (10). So each character can be one of 62 total options. Repeating is okay, so there are (62^5) possible 5-char codes. There are also dragons with 4-char codes, so using the current system for as long as possible, there can be a maximum of (62^5) + (62^4) = 930909168 dragons on DC. Also keep in mind, that's the maximum number of dragons we can have with the current system; a maximum we haven't yet reached.

 

...that's nowhere near enough. Let's see if we can do better.

 

Now let's say we want a 100-gen even gen dragon. We need 1267650600228229401496703205376 CBs for that, which can be shortened to roughly 1.268x10^31. For this problem, I'm going to use the rough completely made-up estimate that out of all the dragons on DC, one third of them are CB. Would six characters be enough?

 

(62^6) + (62^5) + (62^4) = 57731144752. Not even close, and that's before taking into account how many of them are CB. So how long exactly would the codes have to be? Well, long story short, codes will need to be pretty darned long... to be precise, 18 characters long.

 

For those of you who want to see it worked out, that's:

((62^18) + (62^17) + (62^16) + (62^15) + (62^14) + (62^13) + (62^12) + (62^11) + (62^10) + (62^9) + (62^8) + (62^7) + (62^6) + (62^5) + (62^4)) / 3

which works out to

6.2085618x10^31

CB dragons.

 

So 18 characters, finally, are long enough.

 

TL;DR: For those of you looking to breed that epic 100-gen even gen dragon, don't hold your breath. And FORGET about making it PB.

 

 

 

Aren't you so glad you asked?

Edited by Engiype

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List for gens (non-inbred)CB Even Gen:

 

Thanks to Herk I now have formulas you can use for even gen formulas. To find out how many CBs you need 2^n - 1 where n equals the number of generations.

 

2nd gens: 2

3rd gens: 4

4th gens: 8

5th gens: 16

6th gens: 32

7th gens: 64

8th gens: 128

9th gens: 256

10th gen: 512

11th gen: 1024

12th gen: 2048

~~

22 gen: 2097152

~~

51st gen: 1125899906842624 CBs

 

Stair case list for gens (non-inbred) CBs

100 gens: 100

Edited by brairtrainer

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what is the square root of a purple dragon?

 

Oh ho hoh, you're a funny one. Bet you didn't expect me to actually solve this did ya.

 

Well the first thing I did was took the image of a purple dragon and opened it in Paint. I selected it all and shrunk it down to 1 pixel by 1 pixel. This basically takes the average of all the colors in the dragon. Now I have something consistent to work with.

 

I selected that one pixel with the color picker tool, and then I clicked the "edit colors" button. According to Paint, that average color is this:

Red=174

Green=90

Blue=192

 

user posted image

 

It's a relatively simple matter to get the square root of each of those numbers. Rounding each of them to the nearest whole number, we get new values:

Red=13

Green=9

Blue=14

 

When we enter those new numbers into the boxes, it gives us a new hue.

 

user posted image

 

It's not technically pure black, but it's also not very exciting, is it. Just for the heck of it, I'm going to multiply each of those numbers by 10. The proportions are the same, so it should be the same color, only lighter. Here it is:

 

user posted image

 

Hey, you asked.

Edited by Engiype

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Very mind boggling. tongue.gif

 

And if by some miracle, it becomes possible to create a 100 EG, how long will it take?

 

curious.....

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lol well technically u could make a 100 gen dragon with far less using intentional inbreeding xd.png figure out that math will ya lol

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Very mind boggling. tongue.gif

 

And if by some miracle, it becomes possible to create a 100 EG, how long will it take?

 

curious.....

Assuming that you are working in parallel across a sufficient number of players to completely parallelize the process, and all of them have incubate, it would take 5 (days to grow up) * 99 (number of generations excluding the first, for which you do not have to wait) = 495 days = 1 year, 130 days.

 

It only gets longer from there.

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lol well technically u could make a 100 gen dragon with far less using intentional inbreeding xd.png figure out that math will ya lol

Keeping it even gen but also perfectly inbred, you would need just 199 dragons to do that--a mere two per each of 99 generations, plus one (the dragon itself).

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Assuming that you are working in parallel across a sufficient number of players to completely parallelize the process, and all of them have incubate, it would take 5 (days to grow up) * 99 (number of generations excluding the first, for which you do not have to wait) = 495 days = 1 year, 130 days.

 

It only gets longer from there.

It gets a lot longer if there's only one player (theoretically) doing it, ESPECIALLY when you take into account two things:

 

1. Breeding cooldown

2. Egg limits (including moving up trophy levels!)

 

But since there are so many dragons that need to be bred, it would be easy to time it so the first issue is never actually an issue at all. The second one, however, would slow you down. A lot. Brairtrainer's taking this one.

Edited by Engiype

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It gets a lot longer if there's only one player doing it, ESPECIALLY when you take into account two things:

Oh yes. According to what information we have, there are insufficient players to completely parallelize the process anyways.

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OMG you're making my head hurt! wacko.gif

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Oh my god, you guys. XD

 

I actually did wonder how many dragons it would take once (I had seen PonyTales' pink PB project and was like "Oooh! Impressive! I wonder...") but never bothered to figure it out. x3

 

... You crusher of dreams. =P

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Omg, this is very interesting but it's making my head hurt at the same time. Someone actually answered the square root of purple dragon, ROFL. laugh.gif

 

Well, I guess it would only be possible with inbreeding, and still it would take a lot of time. :0

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So lets assume a single player decided to take this epic adventure all by himself.

Lets assume the following variables are true;

- Sufficient number of Incubates (reds)

- NO refusals

- Dragons always give egg

- Eggs gender correctly every time.

 

How long would it take for a player to get to gen 100?

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So lets assume a single player decided to take this epic adventure all by himself.

Lets assume the following variables are true;

- Sufficient number of Incubates (reds)

- NO refusals

- Dragons always give egg

- Eggs gender correctly every time.

 

How long would it take for a player to get to gen 100?

 

Okay, here's my solution.

 

So if we have 1267650600228229401496703205376 CB dragons, we'll have to breed 633825300114114700748351602688 (half of 1267650600228229401496703205376) pairs the first generation to produce 633825300114114700748351602688 eggs. The next generation will be half of that, and so on, until we're down to one dragon.

 

So we need to find out how many pairs we'd need to breed total.

633825300114114700748351602688 + 633825300114114700748351602688/2 + 633825300114114700748351602688/4 ... + 4 + 2 + 1 = ?

 

There's a handy trick, thankfully, for adding up powers of 2. Say you want to add up the powers of 2, up to 2^5. 1 (or 2^0) + 2 + 4 + 8 + 16 + 32 = 63. The shortcut is to do (2^(5+1))-1, which is also 63. So the general formula is (2^(n+1)) - 1. Now we can quickly calculate that we will be breeding 1267650600228229401496703205375 pairs total.

 

Now on to the trickier part. (Oh yes, there's a trickier part.) EGG LIMITS. Since we're starting out with 1267650600228229401496703205376 dragons, we obviously have a gold trophy. So we can deal with 7 pairs at a time before we're egg locked. Then we have to wait 3 days for those eggs to hatch and we can do 7 more.

 

Now towards the end of the line there are fewer dragons we have in the youngest generation than egg slots. What I mean is that when we get to the generation where we have to breed 4 pairs, we can't do 7 at a time any more, because we have to wait for the newest ones to grow up to adulthood (rather than 3 days to hatch) before we can breed them.

 

I'll go to the trouble of dealing with these pesky generations first. From the generation where we have to breed 8 pairs on, it will take 33 days to breed the rest of the dragons.

 

Now we only have 1267650600228229401496703205360 pairs to go! The rest of these are simple to do. Group of 7, then wait 3 days to unlock. Group of 7, wait 3 days. And so on.

 

Between the easy pairs and the pesky ones I did for you, we get a final answer of 543278828669241172070015659473 days.

 

Or 1488435147039016909780864820.474 years.

Or 1488435147039016909780864.820474 millenia.

Or 106316796217072636.41291891574814 times the age of the universe (14 billion years).

 

What's that, you say? You want infinite Incubates?

 

Well that certainly makes it go faster. Using the same technique as before, we get a final answer of:

362185885779494114713343772988 days.

Or 992290098026011273187243213.66575 years.

Or 992290098026011273187243.21366575 millenia.

Or 70877864144715090.941945943833268 times the age of the universe.

 

It'll be a while.

Edited by Engiype

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Now I have an interesting question... or maybe two.

 

How many red dragons would be needed?

 

And assuming whoever attempts it starts with only 1 red egg. How long would it take to get the required amount of reds?

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I am loving this thread. I don't like math enough to work this stuff out, but I LOVE reading what you guys have done.

 

Square root of a purple dragon? Perfect! laugh.gif

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Now I have an interesting question... or maybe two.

 

How many red dragons would be needed?

 

And assuming whoever attempts it starts with only 1 red egg. How long would it take to get the required amount of reds?

Not as long as it could be if you make your Epic 100 gens all Red. wink.gif

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@DragonNighthowler

 

For a perfectly inbred dragon (meaning each breeding pair are siblings aside from the originating CB pair), assuming there's always an egg and it always genders correctly and nobody refuses and you always have enough incubates...

 

First 2 CB dragons (caught at same time) grow up in 2(egg) + 3(hatchling) days = 5 days.

Second generation takes 7(first breeding cooldown) + 2(egg) + 3(hatchling) days = 12 days.

Each subsequent generation will repeat the 12 days until the final 100th generation.

The 100th generation dragon grows up in 2(egg) + 3(hatchtling) days = 5 days.

 

So...

100 generation dragon

1st Generation = 5 days

2-99 Generation = 12 days each for 98 generations.

100th Generation = 5 days.

 

12 * 98 + 5 * 2 = 1186 days = 3 years and 91 days.

(This is of course assuming that one of those years or days isn't on leapyear).

 

The total project would yield 99 generations with 2 dragons each and 1 generation (the last one) with 1 dragon, so you'd have 199 dragons on your scroll dedicated to the project alone.

 

Assuming one person working the project themselves, this is the fastest that project could be completed, since it minimizes the number of dragons growing, meaning that the days to grow up becomes the limiting factor rather than the scroll limit.

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For a non-inbred dragon, assuming there's always an egg and it always genders correctly and nobody refuses and you always have enough incubates, and there are as many people working on this project as necessary to make it take the least amount of time possible...

Also, for ease of math I'm going to assume that everyone has a SILVER trophy (hence, can have 6 eggs at a time) and that all their eggs are going towards the project.

 

100th generation - 1 egg

99th generation - 2 eggs

98th generation - 4 eggs

 

Etc, etc, etc.

 

So the formula for the number of eggs per generation would be 2^(100-n) where n = the current generation.

 

This means 633825300114114700748351602688 dragons will be required in the first generation.

This would be 316912650057057350374175801344 dragon pairs, or going back to the silver trophy assumption, 105637550019019116791391933781 full-time participants required for the first generation, plus one participant contributing a single pair of CB dragons.

 

Hence, DragCave will not be able to do a project on this scale in the as-fast-as-possible mode for a long, long time.

 

That said, the amount of time it would take to complete this project would be significantly faster than the inbred project, since each generation would take 5 days to grow to adulthood.

 

5 * 100 = 500 days, or 1 year and 135 days.

This, again, is assuming there is no leapyear in that timeframe.

Much faster than the inbred project, but much less feasible since we don't have enough users to accomplish this timeframe.

 

Also, this is assuming the project doesn't count as complete until the 100th generation dragon is an adult. (Same assumption is true on my perfectly-inbred estimation).

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Sufficient number of incubates has been previously worked out; for a gold trophy it is 49 smile.gif

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