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Hey guys, this is my first blog post.  As I'm still a noob at this, please tell me if there's anything I need to improve on.

There are tons of blogs about game opinions already, so I figured that I would instead make my blog posts as "guides" - in a way - that examines various aspects of the game mathematically.  Unlike most other games I've played, there doesn't seem to be much theorizing on Valkyrie Crusade despite everything being rather simple math formulas, so I'm hoping to fill in this gap with my periodic silliness about number crunching.

So, for my first blog, I thought I'd expand on something a bit more simple that everyone knows about - buff cards:

Introduction

The meta in Valkyrie Crusade nowadays is reaching the damage cap of 999,999; whereas people used to depend on the +900% buff of Lilim and healing cards to deal enough damage, with buffs being multiplicative and UR team buffers giving +200% or more (and unleashers), we've moved on from that to stacking multiple buffs until we max out our damage - then releasing it all in one massive salvo.

The way that active buffs stack are simple.  A card that gives +200% damage when procced (like Huang Long) will give you a damage multiplier of x3 - so if you were doing 20,000 damage before, you would now do 60,000.  If you add another buff that gives +200% damage on top of this, the two damage multipliers, well, multiply with each other (3x3) giving us a final multiplier of x9, or +800% damage.  "Now, you're doing 180,000 damage.  This means that even small buff amounts snowball quickly when applied multiple times to a team.

All of this is stuff that you've probably already know, so let's get to the point.  We want to figure out how many procs of different buff values we need to reach damage cap.  We also want to consider passive buffs - how much of it is available in-game right now, and how much of it you need to ensure that you need less procs of your buff to reach this damage cap.

Passive Buffs

Passive buffs work a bit differently than active buffs - they stack additively with each other, but multiplicatively with active buffs.  So, if you have three buildings that give +20% attack each, they add up to a total of +60% attack.  If you then use a Summer Alchemist to give your party a +250% buff, your total bonuses are (3.5 x 1.6 - 1) = +460% additional damage..

Without considering cards that add additional passive bonuses as auto-skills (or upgrade potions) or the weird specialized team bonuses, the total amount of passive buffs that everyone can get is 100% (60% Buildings, 10% Element Flush, 25% GUR/GSR only, 5% Friendship).  In the following sections, I'll list some of the more powerful passives, as well as some specialized buffs not available anywhere else.


Player Expected Total Passive Buffs
Beginner 11% (Friendship + Level 1 Buildings)
Experienced 31% (Friendship + Element + Level 4 Buildings x2)
Max Lvl Kingdom

100% (Fr. + Ele. + God. R. Frc. + L10 Buildings x3)

+1 Oracle Chair 150%
+1 Celestial Oracle 200%
+2 Celestial Oracles 300%
+2 Celestial Oracles, Oracle Chair 350%


Friendship

Friendship-icon

There isn't much to say for the friendship bonus.  This is a +5% bonus that pretty much everyone has easy access to no matter which stage of the game you're on.


Buildings (Fire Ruins, Ice Tower, Shrine, Pagan Hall)

  Buff buildings are relatively expensive structures.  At the beginning, a player might only have these at level 1-3 for a total of 6-18% bonus to their attack and defense.  A late-game player can max out all of these buildings for a sweet +60% bonus.  However, as later upgrades of buildings cost upwards of 800,000 in every resource, maxed out buildings are mainly for established players with a lot of resources to burn.


Unit Bonuses

There are two unit bonuses we can get while still maintaining a competent team.  The five-card elemental bonuses (achieved by fielding a team with the same element) gives you a +10% to your cards' attack, which is a significant boost if you are able to get it.  Most of the time, beginning players will have mixed-element teams cobbled together from various events, so this bonus is not really feasible until one has enough cards to create elemental decks.

The other bonus is Godlike Rare Force, a nice +25% for having a team of 5 GUR/GSR cards.  This, too, is only achieved end-game for established players, as orbs are rare and it's difficult to awaken 5 cards when you do not have much resources. 


Awoken Burst (+100% ~ +200% ATT)

Awoken Burst is an effect that activates only on awakened SR and UR cards. When active, it gives a +100% passive bonus to attack (and soldier count) for GUR cards and a +200% passive bonus for GSR cards. Burst is notable for breaking the damage limit on Legendary Archwitch fights, dealing upwards of 4 million damage with elemental advantage.

Even though it is a randomly activated effect, Awoken Burst is considered a passive effect. It stacks additively with your other passive effects like buildings and friendship, which means that you're usually not getting double your damage during it. The 4 million max damage is significant, however, and another section will be dedicated to this effect.


Archwitch Damage Bonus (+600% ~ +1400% ATT)

There are cash-only cards, released during every event period, that carries a secondary effect like:

Deal ??X DMG to Archwitches / Subjugation Points Earned ???% UP Only available against current Archwitches.

The multiplier and subjugation bonus varies with the card and evolve level, but the damage bonus are always considered passive effects and are additive with other such bonuses.


Celestial Oracle (Event-Only, +100% ATT)

Celestial Oracle H

The Oracle, in all of her OP-ness.

Celestial Oracle is a game-changer card.  As an unleasher (welcomed on most every team) that also sports a passive +100% attack bonus, Celestial Oracle single-handedly reduces the number of buffs needed to reduce the number of buffs you need for max damage.  She was present back when MyNet was nicer and gave out amalgation cards in the event progression (rather than making them rank-only then inflating the ranks to be impossible for new players).

You are limited to a maximum of 2 Celestial Oracles per deck due to only two laurels being available per person, but Oracle Chair - the LAW and pre-amalgated form of Super-OP Oracle - is not shabby either with her +50% passive attack bonus.  If you have this card and are a few percentages short from being able to max damage with one less buff, heavily consider adding her instead of, say, another copy of Arpa.

Of course, for the rest of us who weren't present for this event, this card is pretty much a pipe dream.


Krazy Kitty Kat (ABB-Only, +150% ATT/DEF)

Krazy Kitty Kat H

Krazy Kitty Kat

Krazy Kitty Kat is an ABB reward with an unfortunate abbreviation released around June 2016.  As an attack buffer, it leaves a bit to be desired (with an attack buff of only +250% and no defense buff), but comes with an amazing secondary passive of +150% to ATT and DEF - higher than even Celestial Oracle!

Unfortunately, if you use Krazy Kitty Kat, it's probably as your primary buffer.  Kat's +250% attack buff is pretty nice, but even with the passive in place, the lack of a defense buff does hurt when you are fighting against strong enemies like LAWs.  On top of that, the amalgated form of Kat is not really a free card; ABB ranking pretty much requires you to use rods (even in the top 1000), no matter how active you are.

We'll take another look at the Kat later on, in the buffs section.


Active Buffs

Formulas

Let's get to the meat of this blog article.  Emperical testing has shown us that you need approximately 1.4 million ATK in order to reliably deal 999,999 damage with a normal attack.  If this is what we are aiming for, then using a GUR card with maxed attack (39,999), we will need a final multiplier of x35 (+3400% damage) in order to achieve this.

In order to calculate how many procs a buffer needs to achieve this x35 multiplier, we must thus solve the following equation for the lowest N:

(BuffAmount + 100%) ^ N >= 35

Solving this gives us the following equation:

N >= ln(35) / ln(BuffAmount + 100%)

We can use this to get the tables below.


The 35x Multiplier

As mentioned before, the 35x multiplier is where our damage reliably caps at 999,999, and is where we want to aim for everything except for salvo cards (and awaken burst with LAWs).  With each of the buff amounts currently existing in the game (excluding the +900% from Lilim), we arrive at the following:

Passive Bonus Needed for Max Damage on N Procs
Buff % # Base Procs 1 2 3 4 5 6
+650% 2 +367% +0%
+500% 2 +483% +0%
+400% 3 +600% +40% +0%
+300% 3 +775% +119% +0%
+250% 3 +900% +186% +0%
+200% 4 +1067% +289% +30% +0%
+150% 4 +1300% +460% +124% +0%
+100% 6 +1650% +775% +338% +119% +9% +0%

For this table, the first row is the buff amount, and the second row is the # of buff procs needed to achieve 999,999 damage without any extra passives.  The next 6 rows show the amount of passive bonuses you need to achieve a desired number of procs

Let's use Baal as an example.  Baal's attack buff is +400%, which means that according to the table she needs 3 procs normally to have your attacker deal capped damage.  If you want her to do this with 2 procs instead, you'll need a total of +40% in passive bonuses, which is a relatively easy task to accomplish.  On the other hand, Nadeshiko's attack buff is +300%.  She also needs 3 procs normally, but in order to reduce this to 2, she needs a combined passive buff of +119%.  You cannot accomplish this with just buildings and unit bonuses alone - you need either an upgrade potion or a (Autoskill) attack passive to do this!

As we expected, cards with a 500% attack buff or higher will achieve maximum damage in two procs.  A card with +400% attack buff might as well achieve maximum damage in two procs, falling off only if you are still a beginner and don't have the requisite passive bonuses yet.  A card with +300%, +250%, and +200% perform similarly at 3 procs (the +200% buffer needs a miniscule +30% in passive buffs to do so), with the +300% buffers potentially reaching the 2 procs threshold with a oracle chain and a high-level kingdom.


The 70x Multiplier

Why is the 70x Multiplier important?  Well, this is where 10x hit cards also reach 999,999 damage, and is needed if we want to maximize our damage with cards like Halloween Hades.  The 70x multipler is much harder to hit compared to the 35x multipler, and you'll notice that a lot of buffers struggle at this threshold:

Passive Bonus Needed for Max Damage on N Procs
Buff % # Base Procs 1 2 3 4 5 6
+650% 3 +833% +24% +0%
+500% 3 +1067% +94% +0%
+400% 3 +1300% +180% +0%
+300% 4 +1650% +338% +9% +0%
+250% 4 +1900% +471% +63% +0%
+200% 4 +2233% +678% +159% +0%
+150% 5 +2700% +1020% +348% +79% +0%
+100% 7 +3400% +1650% +775% +338% +119% +9%

+650% buff cards are still extremely strong, and can easily buff a Halloween Hades to max damage in 2 procs with a mimimal 24% passive.  A +500% buff card needs a +94% passive to do the same, which means an almost maxed kingdom or Celestial Oracles need to be added.

+400%, +300%, and +250% cards are still strong, able to buff a Halloween Hades to max damage in 3 procs (with some passives needed on the latter two).  The bog-standard Huang Long/Elemental Queen (+200%) buff card suffer quite a bit here, needing 4 procs to max out HH's damage wihout powerful cards like Celestial Oracle.

The difference between a +250% card (Summer Alchemist) and a +200% card (Huang Long) is pretty pronounced here.  With just kingdom buffs, Summer Alchemist is able to max out HH's damage in 3 procs, while Huang Long can only do it in 4 without Celestial Oracle's help.


The 140x Multiplier

The 140x multiplier is exclusively for maximizing damage during Legendary Archwitches.  It allows you to deal 4 million damage during awakened burst, which helps take down these end-game bosses that much faster:

Passive Bonus Needed for Max Damage on N Procs
Buff % # Base Procs 1 2 3 4 5 6
+650% 3 +1667% +49% +0%
+500% 3 +2133% +189% +0%
+400% 3 +2600% +360% +0%
+300% 4 +3300% +675% +19% +0%
+250% 4 +3800% +943% +127% +0%
+200% 4 +4467% +1356% +319% +0%
+150% 5 +5400% +2040% +696% +158% +0%
+100% 7 +6800% +3300% +1550% +675% +238% +19%

This table has the +100% passive from Awoken Burst already factored in.

Here, the godly +650% buffers shows their superiority by maxing out awoken burst in just 2 procs with any decent bonuses from your castle.  500% buffers have considerably more trouble with this, and require at least one celestial oracle (or some other +89% ATK passive card) to be able to achieve the same.

+400% and +300% buffers perform similarly here.  Both are able to max out awoken burst with 3 procs, with the +300% card requiring minimal assistance from your kingdom to do so.

+250% and +200% buffers max out awoken burst with 4 procs.  A +250% buffer can shorten this to 3 procs with a Oracle Chair (or other passive ATK) card, but is unable to do this with just your kingdom buffs.  Still, a +250% buffer gets pretty close even with just 3 procs, doing 3.67 million damage instead.

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