Identifying the Optimal Quanta Index

[Thurhame]

I. Introduction

Most of you are probably familiar with Scaredgirl's Quanta Index formula. It is a very useful number for figuring out about how many pillars/towers a deck should have. For those who aren't familiar with it, here is a summary:

Quanta Index (QI) = (total cost of all cards + ability costs - total quanta produced by spells - ability quanta) / (number of pillars in deck)

However, there is much talk about how the ideal QI varies depending on various factors, most especially the "speed" of the deck. I have looked for some way to quantify this variation, but found none, so (being a math nerd) I decided to calculate it myself.

II. The Formula

optimal QI for non-Mark elements = 1/4 * (T-1) * ( (T+12)/(T+6) + (T+14)/(T+7) )

optimal QI for the Mark's element = (non-Mark formula) + 1/2 * (T-1) * ( 1/(T+6) + 1/(T+7) ) * (number of cards in deck) / (number of pillars in deck)

*where T is the deck's average # of turns to win a game

III. The Theory

Also known as, "How this crazy guy came up with this weird-looking formula."

This formula uses the idea of Scaredgirl's ideal situation, where on average you produce exactly enough quanta to play all the cards you draw. Since cost increases linearly as the game progresses, while pillar quanta increases quadratically, this ideal situation occurs at a particular length of time. Obviously, we want this length of time to coincide with the length of the game. This situation can be described by an equation, which simplifies to the above formula. The equation varies slightly depending on whether the deck takes the first or second turn; I took the average of the results for this formula.

The original equation, which assumes you go first and ignores the Mark, was:

SUM[k=7..N-1]{kP/D} = QN/D

*where N is the number of cards drawn (i.e. T+6), P is the number of pillars in the deck, D is the number of cards in the deck, and Q is the numerator of the QI formula

Endnotes
1. Remember to use QI SEPARATELY for each element.
2. A tower counts as both a pillar and a spell that produces 1 quantum.
3. A quantum pillar counts as 1/4 pillar of each element. A pendulum counts as 1/2 pillar of both elements.
4. The quanta produced by Immolation/Cremation should be reduced based on T and the number of creatures in the deck you are willing to sacrifice.
5. Cards such as Fractal do not receive any special treatment.
6. There is much disagreement about how to count abilities. I prefer to use Scaredgirl's original method, but how you figure it is up to you.
7. I have no idea how to handle hastened card drawing. If you do, please share it.
8. The variable T in my formula is often referred to as "TTK," which stands for "Turns To Kill." It is commonly used to measure a deck's "speed."
9. If I have made a mistake somewhere in my formula, please let me know. Also, comments and criticism are welcome.

[rosutosefi]

I've done some calculations for this, too. One important thing to note is the deck's "launch time". This would be the time that you drop all the cards from your hand with sufficient quanta production to fuel the next cards. This would mean that a deck with many small critters need only a few towers because less quanta is needed to fuel future cards you draw. On the flip side, decks with dragons and other high-cost cards should have a larger starting amount of quanta to fuel for this launch, while leaving a large amount of quantum late-game. I tried this already, and found that it is not TTW alone that dictates the optimal amount of quanta you need, but also the launch time, which means you have to account the average quanta needed for the cards you use, and card playing priority, which means that you need a relatively lower QI if the cards you need to play early game have a relatively low cost and the opposite of that when you need to play high cost cards early.

I'll go look for my notes about that. 

EDIT: Here you go. Yes, I quit on that thing.

[bogtro]

Without verifying the formulae too much, the main problem that I see is that TTW and QI are directly related and therefore you can't really get one from the other. Changing the QI changes the TTW, leaving an infinite loop.

[rosutosefi]

Without verifying the formulae too much, the main problem that I see is that TTW and QI are directly related and therefore you can't really get one from the other. Changing the QI changes the TTW, leaving an infinite loop.

That's why you make a program for it.

[Baubble]

I have a question about the setup of the original formula

Quanta Index (QI) = (total cost of all cards + ability costs - total quanta produced by spells - ability quanta) / (number of pillars in deck)

Would total quanta produced by spells and ability quanta (I'm assuming like Devourer or RoL, etc) be on the top line, or would they be on the bottom line?  Or wouldn't it matter?
I ask because I'm a humanities major, not a math major. :-(

[rosutosefi]

I have a question about the setup of the original formula

Quanta Index (QI) = (total cost of all cards + ability costs - total quanta produced by spells - ability quanta) / (number of pillars in deck)

Would total quanta produced by spells and ability quanta (I'm assuming like Devourer or RoL, etc) be on the top line, or would they be on the bottom line?  Or wouldn't it matter?
I ask because I'm a humanities major, not a math major. :-(

That is also one question I have, since quanta-producing creatures also act as pillars and are also affected by "launch". And ability costs should have some kind of multiplier depending on how often that skill is used. For example, one would spam growth or scarab but not lobotomize or devour.

For now, I just assume devourers and RoL's are played immediately. Include the devourer's quanta cost, but treat it also as a pillar.

[eaglgenes101]

Some1 stick this into a tool?

[Baubble]

That is also one question I have, since quanta-producing creatures also act as pillars and are also affected by "launch". And ability costs should have some kind of multiplier depending on how often that skill is used. For example, one would spam growth or scarab but not lobotomize or devour.

For now, I just assume devourers and RoL's are played immediately. Include the devourer's quanta cost, but treat it also as a pillar.

ScaredGirl addressed ability counts in her original post.  I have no idea on the efficacy, but it has worked for me in the past.  Abilities that are intended to be played every turn have their quanta costs counted twice, while situation-specific abilities are counted once.

Not sure how this would change when you factor launch in. 
I imagine that abilities with costs would be calculated independently since you're already counting for the speed of the card, so you'd be accounting for it twice otherwise.

[shileka]

Spoiler for Hidden:

I. Introduction

Most of you are probably familiar with Scaredgirl's Quanta Index formula. It is a very useful number for figuring out about how many pillars/towers a deck should have. For those who aren't familiar with it, here is a summary:

Quanta Index (QI) = (total cost of all cards + ability costs - total quanta produced by spells - ability quanta) / (number of pillars in deck)

However, there is much talk about how the ideal QI varies depending on various factors, most especially the "speed" of the deck. I have looked for some way to quantify this variation, but found none, so (being a math nerd) I decided to calculate it myself.

II. The Formula

optimal QI for non-Mark elements = 1/4 * (T-1) * ( (T+12)/(T+6) + (T+14)/(T+7) )

optimal QI for the Mark's element = (non-Mark formula) + 1/2 * (T-1) * ( 1/(T+6) + 1/(T+7) ) * (number of cards in deck) / (number of pillars in deck)

*where T is the deck's average # of turns to win a game

III. The Theory

Also known as, "How this crazy guy came up with this weird-looking formula."

This formula uses the idea of Scaredgirl's ideal situation, where on average you produce exactly enough quanta to play all the cards you draw. Since cost increases linearly as the game progresses, while pillar quanta increases quadratically, this ideal situation occurs at a particular length of time. Obviously, we want this length of time to coincide with the length of the game. This situation can be described by an equation, which simplifies to the above formula. The equation varies slightly depending on whether the deck takes the first or second turn; I took the average of the results for this formula.

The original equation, which assumes you go first and ignores the Mark, was:

SUM[k=7..N-1]{kP/D} = QN/D

*where N is the number of cards drawn (i.e. T+6), P is the number of pillars in the deck, D is the number of cards in the deck, and Q is the numerator of the QI formula

Endnotes
1. Remember to use QI SEPARATELY for each element.
2. A tower counts as both a pillar and a spell that produces 1 quantum.
3. A quantum pillar counts as 1/4 pillar of each element. A pendulum counts as 1/2 pillar of both elements.
4. The quanta produced by Immolation/Cremation should be reduced based on T and the number of creatures in the deck you are willing to sacrifice.
5. Cards such as Fractal do not receive any special treatment.
6. There is much disagreement about how to count abilities. I prefer to use Scaredgirl's original method, but how you figure it is up to you.
7. I have no idea how to handle hastened card drawing. If you do, please share it.
8. The variable T in my formula is often referred to as "TTK," which stands for "Turns To Kill." It is commonly used to measure a deck's "speed."
9. If I have made a mistake somewhere in my formula, please let me know. Also, comments and criticism are welcome.

okay, i planned to learn how to calculate the QI with my head, but now i see thats gona be a whole lot harder than expected...

[Acsabi44]

okay, i planned to learn how to calculate the QI with my head, but now i see thats gona be a whole lot harder than expected...

Lol, it is actually very easy.
Here's the basic forumla that works for 80% of the decks:
1, slap 12 pillars into that deck.
2, play lots of games.
3, adjust. (most probably it will be in the +-2 range)
Believe me, it will work for you

The remaining 20%? You don't wanna run those kind of decks until you are not adept in determinig proper pillar amounts by playing a few games.

[shileka]

okay, i planned to learn how to calculate the QI with my head, but now i see thats gona be a whole lot harder than expected...

Lol, it is actually very easy.
Here's the basic forumla that works for 80% of the decks:
1, slap 12 pillars into that deck.
2, play lots of games.
3, adjust. (most probably it will be in the +-2 range)
Believe me, it will work for you

The remaining 20%? You don't wanna run those kind of decks until you are not adept in determinig proper pillar amounts by playing a few games.

i meant, that i want to be able to see if a deck has enough quanta just by seeing the cards, no testing, like a QI calculator in my head

[Shantu]

That comes with experience. QI isn't very reliable.