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[Guide] Numerical Analysis of TOS {WIP}
#1
TABLE OF CONTENT
  1. Introduction
  2. Foundations
    1. How to Read DSI Tables
    2. Damage Calculation
    3. Healing Calculation
    4. Damage Reduction
  3. Strategy Planning and Execution
    1. Stage Analysis
    2. Stalling and Bursting
    3. Rune Spinning and Stacking
    4. Diamond Usage
    5. Case Handling In a Pinch
  4. Resources Management
    1. Level
    2. Skill Level
    3. Amelioration
    4. Resources Distribution
  5. Miscellaneous
    1. Seal Card Drawing
1. INTRODUCTION

Hello everyone, I am a fellow TOS player. Being a close to F2P player (6 dimes top up for the new dragons), I personally never skip a single SM/NM from my second month of playing onward. I deliberately skip Saruman and Lunar Sirius which are too early for me and Solar sirius is my first SM which I clear using 2 dimes.

So, how I do so? How I come up with strategies to face different kind of stages? Being an almost F2P player, how I manage my resources? All these have to do with mathematics. I plan my strategy and do the calculation before I ever attempt the stage so that I know exactly what to do like how many combo/runes I need to dissolve to kill the enemy, when to use my skill, when should I stall etc. With all these planning, even being restricted by resources as a F2P player, I still can find my own way to attempt stages based on what I have in my inventory.

Therefore, I am going to compile my knowledge as a guide.

2. FOUNDATIONS

First of all, I am going to teach your guys the necessary knowledge and mathematical equations required for strategy planning.

2.1. HOW TO READ DSI TABLES

In order to plan any strategy, it is important to learn about your enemy first. In forum, we have DSI sections whereby OP will list out the details of enemies in a stage in table form. All new DSI should be done in table form while some older DSI does not as forum does not have table function supported last time.

A general DSI table should have the following format which is similar to what we have in wiki.
WaveEnemyATKCDHPDEFSkill EffectNote
Wave No.Enemy NamexCD(cd)yzDescription of enemy skillsAnything not from skills but worth mentioning
Wave
Indicates which round the enemy will appear. If it is left empty, it means that it is a mob whereby it appears in any round that is not mentioned in the subsequent rows.
Enemy 
Usually just the enemy's name but some hardworking poster will use picture / coloured font which will tell you the attribute of enemy.
ATK 
Indicates the base damage of enemy. For certain enemy with trojan/berserk/hp reduction skills, please refer to skill description
CD
As you can see in the table above, there are two values, one is within bracket and one doesn't. The value within bracket is what we called initial CD that is the enemy's CD when you enter the stage. The other value is the CD the enemy will become every time it makes an attack. So, an enemy with CD 2 (5) means he will have CD of 5 every time you enter the stage but after he make his first attack, he will attack again after every alternate turn (CD2).
For enemy without the value in bracket, it means that its initial CD is within x+-x/2 whereby x/2 is the integer division. For example, for a monster with CD3, its initial CD can range from 2(=3-1) to 4(=3+1)  while a monster with CD4 can have initial CD range from 2(=4-2) to 6(=4+2).
Another important fact about CD is for enemies with two hp bars. The initial CD of the enemy in 2nd hp form is directly inherited from the 1st hp bar when it is killed.
HP
It tells you the enemy's hp which will decide how much damage you need to deal in order to kill him.
DEF
It tells you the defense value of the enemy, which will be deducted from your damage to get the actual damage. Usually, the value will be ignored as it does not affect your damage much. But there are enemies with really high defense which can be an annoyance. The detailed calculation will be covered in the next section.
Skill Description
The most important sections in a DSI table. It tells you the skill effect of the enemy which sometimes can be really sadistic and most of the strategy planning are done to counteract the enemies' skill. Please read the description carefully so that you won't get screwed by it.
Note
Used for any miscellaneous information like drops, raid cards, 2 hp bar etc

2.2. DAMAGE CALCULATION

The primary objective of any stages in this game is to kill your enemy. That's why it is important to be able to know how much damage you can deal and the way to do it is via mathematical formula.

First of all, if you are observant enough, you should know that a value will be shown on each card's icon. That's your card damage before it undergoes any modification and I call that your base damage. 

Base Damage = Base Attack x Rune Factor x Combo Factor x Skill Multiplier x Amelioration x Dragonic Compulsion
I know that the formula is rather long but that's what happens in the game's mechanics. However, in general, we can ignore amelioration and dragonic compulsion in our calculation unless you are looking for a super massive burst then you would like to include them as they are often used for burst only.

Base Attack
Base Attack is simply the attack of each cards. During a battle, you can hold on any card and you can see the base attack of the cards.
There are multiple ways to increase/decrease a card's base attack other than leveling:
  1. Absorption
    There are only two cards capable to do so in the current version which are Michael Lucifer and Novalis. They deduct the base attack from the team members and add to their own base attack. This is by no mean a damage booster as it does not change the total base attack of team at all. However, when they are paired with Libra or Ursula respectively, it makes their skills much more powerful.
  2. Transfer from Recovery
    There are 4 cards currently capable to do so. The first one is Cassandra whereby her active make your recovery 0 while add 2.5 times of the deducted value to your base attack. The other 3 cards are the PR RGB ED whereby their leader skill add the recovery to your base attack.
  3. Hidden Passive
    When certain cards are used together, we can have additional boost to the base attack. There are too many of them for me to list out.
  4. Fusion
    When two cards fuse together, their stats change to the one based on fusion stats.  
For mono team, since they share the value for all the other factors, we usually use the team total base attack for damage calculation for simplicity. Similarly for ML/Novalis team, since the two cards have significantly much higher base damage than other cards with their actives, we use their base damage for damage calculation. For rainbow team, we can't do so as each attribute have seperate rune factors to consider.

Rune Factor
The value of rune factor depends on the runes you dissolve and each attribute have a seperate rune factor.
Rune Factor for X attribute = (No. of X rune dissolved + No. of group of X runes dissolved)*0.25 + No. of enchanted X rune dissolved * Enchanted Bonus

Lets do some sample calculation.
Lets say for dark rune factor, you dissolve 3 non-enchanted dark rune in 1 group, so your rune factor will (3+1)*0.25 = 1
Now, lets say you have 11 non-enchanted dark runes to dissolve,
if you dissolve as group of 3, 3 and 5, your rune factor will be (11+3)*0.25 = 3.5
if you dissolve as group of 3, 4 and 4, your rune factor will still be (11+3)*0.25 = 3.5
However, if you dissolve them as a single group of 11, your rune factor will be (11+1)*0.25 = 3
So, it is optimal to split your runes to be dissolved as many as possible in order to maximize the rune factor.

Next, lets move to enchanted bonus, for any default team, the value of enchanted bonus will be 0.15. So, each enchanted rune contributes 0.4 to rune factor as compared to non-enchanted rune's 0.25. Therefore, 1 enchanted rune = 1.6 non-enchanted rune.
For a dual Norse team, they have hidden passive whereby the enchanted bonus is increased from 0.1 to 0.25. So, each enchanted rune has rune factor of 0.5 which is twice the amount of non-enchanted rune. Therefore, for a dual Norse team, Norse leader's enchantment is effectively a 2x damage multiplier while it is only 1.4x for a Norse-Daji team.

Combo Factor
If you are new to TOS, you will often see people tends to dissolve as many combo as possible instead of dissolving the runes as one big group. Besides contributing to higher rune factor, it also contributes to higher combo factor.
Combo Factor = 1 + (No. of combo + Extra combo(Greek Only) - 1) * Combo Bonus
Finally, we have some much simpler formula. The only thing I need to explain here is the combo bonus. Combo bonus is the constant value increment of combo factor for every extra combo made. By default, the system set it to 0.25. However, there are a few cards that have the ability to improve it.

  1. PR Daji (The 5*/6* version does similar thing so I only talk about the PR version)
    The most OP leader in the current game. Her leader skill increases combo bonus by 125% which is equal to 1.25. Furthermore, when she is used as leader/ally, the combo bonus is further increased by 25% which is 0.25. So, if you use PR Daji as leader/ally, your combo bonus will be increased from 0.25 to 1.75!
  2. Circe
    The green enchantress which was once very popular as damage booster due to the low risk accompanied by her active. Her active increases combo bonus by 0.25.
  3. Wavel
    Also commonly called as dragon daji which also increases combo bonus by 1.25 restricted to a pure dragon team but without the hidden bonus. 
Since I am at this section, I shall answer a few questions that have been around TOS forum for a very long time.
1) Just how OP PR Daji is as compared to other multiplier leaders?
From first glance, people may think that PR Daji is as effective as a 7x multiplier leader but that only happens at infinite combo. In fact, PR Daji's effective damage multiplication increases as your combo increases.
To compare PR Daji with other leader, I will show the result in table form:
No. of Combo MadeCombo Factor with PR DajiDefault Combo Factor Effective Multiplier
1100%100%1
5800%200%4
91500%300%5
213600%600%6
457800%1200%6.5
Consider most players can make combo in between 5 to 7, PR Daji is effectively a 4 to 4.5x multiplier leader.
2) Since PR Daji is so OP, why don't we use it as both leader and ally?
First of all, we need to understand that Daji increases combo bonus by addition rather than multiplication. Therefore, using Daji as both leader and ally will never increases your combo bonus to twice the combo bonus when you use Daji as leader/ally only and there are so many other leaders which multiply your damage more than 2x.
3) Why people always say don't use Circe together with PR Daji?
The answer is actually very similar to the previous question. Again, Circe also increases combo bonus by addition, so using Circe in a team with PR Daji will only increase combo factor from 1.75 to 2. It is not like they do not stack, but the effective multiplier is less than 1.15 which increases overall damage by less than 15%. Therefore, it is often advisable to use some other booster besides Circe.
4) Why people don't use Circe in Greek Deck while Circe works well with high combo?
This is because MH make Circe's active does not work in greek deck. Blame MH for this.  Angry

Skill Multiplier
There are tons of skills in TOS that can improve your damage by simple multiplication as long as you fulfill the condition. In general, there are 3 kinds of skill multiplier:
  1. Leader Skill
    As long as you fulfill the condition, leader skill will increases your damage by multiplication according to description.
  2. Everlasting Active
    As far as I know, only 1 card go to this category: Medea. They will increases your damage like leader skill forever once you activate them until you die or deactivate them. PR Yan and Xi does not go to this category as they make rune factor's accumulation easier rather than boost your damage. Similar for Cassandra and Circe as they belongs to base damage and combo factor respectively.
  3. Active
    There are tons of them available in TOS. I won't list them out here to save space and time. These skills will place a buff on the left side of the screen and there are be 4 buffs at most at one time.
For your knowledge, there are some skill multipliers which are not constant, I will include the formula for them. For the rest, just follow skill description.

  1. Hades (Leader skill)
    1 + No. of attribute in the team * 0.5
  2. Jyu Yun (Leader skill)
    2 + No. of human in the team * 0.25
  3. PR Nian (Leader skill)
    1 + No. of attribute of beast in the team * 0.5
  4. Horned King (Active)
     1 + No. of attribute in the team * 0.5   (Capped at 4 attribute)
  5. Odin (Active)
    1 + (Max HP - Current HP)/Max HP * 1.5
  6. PR Odin (Active)
    1 + (Max HP - Current HP)/Max HP * 2
Amelioration
If you activates it, just multiply the corresponding value, nothing much to say.

Dragonic Compulsion
In this section, I will only talk about the damage increase from DC.
First, to activate the skills, you need to score
>=2000 for first skill
>=4500 for second skill
>=7000 for third skill
If you activates it, just multiply them to the damage accordingly.
Next, the score itself also contributes as damage multiplier. The formula is

1* craft:

1 + Score/10000 for Score <=5000
1.5 + (Score-5000)/40000 for Score >=5000


2* craft:
1 + Score/10000 for Score <=7000
1.7 + 1.5*(Score-7000)/40000 for Score >=7000


3* craft:
1 + Score/10000 for Score <=10000
2 + (Score-10000)/20000 for Score >=10000
Damage Modification
With the base damage calculated, it is times for damage modification.
There are generally 4 factors that can modify your damage.

  1. Attribute
    If your attack have attribute advantage, multiply it by 1.5.
    Else if it has disadvantage, multiply by 0.5.
    Else, multiply by 1.
  2. Your actives
    Take note that active of certain cards like Ursula, Luna, Hertz does not really fit into the skill multiplier session. For Ursula and Luna, replace the attack of specific members with the member with highest attack. (Often Novalis and ML). For Hertz, you can take it as times 2 but remember to do attribute mod. seperately.
  3. Enemy's Skill
    There are generally 4 kinds of enemy skill that can affect your damage.
    1) Reduction
    We often see enemies with skills like beast shield, water shield, etc. If you do not fulfill the condition, your damage will be divided by a particular factor. For general racist/mono racist shield, the factor is 4. The most extreme I ever seen is water XY extra whereby SSS divides light and dark attack by 20.
    2) Condition
    If you fulfill the condition, do damage without modification. If you fail to do so, damage becomes 1.
    3) Ignore damage
    Skills like initial shield/ambush shield/kamikaze states that they will not receive damage unless the condition is fulfilled. So, your normal damage will becomes 1.
    4) Tumbler shield
    It simply cap your damage based on percentage of enemy's hp.
  4. Defense
    After doing all the modification, we have the final value for our damage. Now, we can just minus off the defense from this value and we get our expected damage. If the resultant value is less than 1, the resultant damage is 1. Take note that I put defense as final modification because it is last in system's order. One good example to illustrate this is Path of Atavism boss with 900k defense and beast shield. If you are using non-beast to attack, it will be divided by 4 followed by minus 900k instead of minus 900k followed by division by 4. The prior one will need to do 3.6mil damage for defense break while the latter one need only 900k.
Simplification of Calculation
I am pretty sure some of you are already taunted by such a long process of calculation.
Don't be afraid, my friends. Even I don't do the calculation like this, only the system do so. What I usually do is like this:
1) Multiply your base attack with leader skill first to get a base value. These values are generally constant so we can reuse them over and over again.
2) Next, assume you need to do 5/7 combo, which gives you the particular combo factor (2 and 2.5 for default team, 8 and 11.5 for team with PR Daji, etc). Again, these values will be reused again and again.
3) Calculate the rune factors. Just remember 3 runes in a group gives you 1 and every additional runes in the group give extra 0.25. Ignore enchanted bonus as we don't mind to deal extra damage unless you are using dual norse whereby enchanted bonus is really significant.
4) Just multiply these 3 together with active's multiplier and apply modification, then you can easily get your expected damage.
Unless you find that you fall short of damage from your calculation, then you need to do the full calculation and add in whatever bonus that you can possibly achieve.
If you are going to apply DC, it is safe to assume it is a 1.5x multiplier since score beyond 5000 contributes very little to damage boost.
So, the key in simplify calculation is always take the lower bound, we don't mind to do extra damage. If your lower bound can actually meet the damage requirement, then you can be very assured that your actual damage will be enough to kill the boss.

As you can see, I did not include sky drop into my calculation. This is because it is very unpredictable. As I take lower bound to simplify calculation, I usually assume that there is no sky drop in my calculation. Therefore, sorry to say, greek's damage will be highly unpredictable as they rely on sky drop and that's also the reason I rarely play greeks as it is very difficult to numerically analyse them.

2.3. HEALING CALCULATION

Healing is another important mechanism in the game as it allows you to stall against enemy. It is often required when you need to stall for skills and not killing your enemy. Luckily, heal formula is similar yet much simpler than damage formula.

Heal = Base Recovery X Rune Factor X Combo Factor X Skill Effect X Amelioration X Dragonic Compulsion
Base Recovery
Similar to base attack, there are a few factors that can affect it:

  1. Cassandra/PR RGB ED
    She makes your base recovery 0, as simple as that.
  2. ED/Yuki-Onna
    Add a constant amount of recovery to specific members.
  3. Enemy Skill
    There are sadistic enemy skill which makes your recovery 0.
Rune Factor
Exactly the same as damage section except that nothing increase the enchantment bonus like dual Norse.

Combo Factor
Exactly the same as damage section except that nothing increase the combo bonus. So, PR Daji does not help you to heal extra except for her insanely high recovery.

Skill Effect
Again, the same as damage section, there are actives and leader skills that multiply your recovery.

Amelioration
Once again, same as damage section.

Dragonic Compulsion
Score does not contribute to recovery multiplier but there are elves craft whereby the skill increases recovery.

There is no additional modification so healing is much simpler to calculate. Just get your base recovery, combo factor, rune factor and skill factors, we are done in most cases.

Besides these healing, there are three more special kind of healing we have.

  1. Leeching
    Exclusive to Ursula, Barbara and Wavel's active. You heal by 5% of damage you deal. Capped at 50% of max hp.
  2. Percentage Healing
    Although PR RGB ED make your base recovery 0, they do provide you another healing mechanism. You heal 2% of loss hp for every dragon members for every heart rune dissolve. So, you heal 36% of loss hp for 3 heart runes if you run a dragon mono.
  3. Dragonic Compulsion
    For certain crafts, one of its skill is heal 50% of your max hp if you have enough score to activate it.

2.4. DAMAGE REDUCTION

Besides healing, another important aspect in stalling is damage reduction as it helps you to reduce damage, therefore you can stay alive longer and stall longer. One important feature in TOS is as long you are alive, you can continue to deal damage. Therefore, the longer the time the enemy takes to kill you, the more chance of attack you have. If you can stall indefinitely, then you can kill the boss no matter how much hp he has (Unless he heals). This is also the reason why we can see a lot of teams with low damage capability can pass stages whereby team with much higher damage can't pass.

So, how does damage reduction works?
There are mainly 2 kinds of damage reduction:
1) Tumbler 
Tumbler works as damage limiter. When you have tumbler in active, when an attack is supposed to kill you, the tumbler effect will make you have 1 hp remained no matter how high your opponent's damage is. Therefore, as long as you can heal back enough hp to activate the tumbler again, you will be immortal. However, there is one catch for tumbler that is it can only be activated once per turn with exception of IPQOW's active 1st turn. Therefore, against multiple enemy that attack together, tumbler is not that useful.

There is no complicated equation for tumbler. When your enemy's attack exceeds your current hp and your current hp is more than a certain percentage, then you will survive with 1 hp.

2) Direct damage reduction
This refers to many leader skills and actives. In general, when we talk about damage reduction leader, we have defensive dragon, certain zodiac, Penetrating Eye Fong Tian-Ling, and our mightly Wukong.
These leaders have leader skill that reduces attack from particular attribute by X%. However, we are not that interested how much damage we actually reduce, but the damage we receive after the reduction.

So, the formula is
Damage Received = Original Damage * (1-X/100)
The formula is kinda simple and most people should be able to derive this even without looking at this guide. However, very often, we have team that use damage reduction leader as both leader and ally. So, given one reduces X% and one reduces Y%, how much damage we receive in the end?
The formula is

Damage Received = Original Damage * (1-X/100) * (1-Y/100)

So, don't go and sum up the 2 percentage because that's totally wrong. Imagine you have 2 defensive dragons that -50%, is it logical enough to have 100% damage reduction? In fact, 2 50% reduction will only gives you 75% reduction as you only receive (1-0.5)(1-0.5)=0.25=25% damage in the end.
By applying these formulas, we can easily get the compound damage reduction and determine whether we can survive the stage or not.
In general, reduction leader is good for stages with 1 or 2 attributes only since their reduction only works for specific element with exception of Wukong and I am going to cover him in the next section.

Mightly Wukong
Okay, this guy deserves a section just for him for two main reasons:
1) This guy has the highest reduction percentage which is 80% after amelioration and his reduction applies to all attributes unlike other reduction leader.
So, how much damage will you receive you run this awesome cards as both leader and ally?
The answer is

(1-80/100)*(1-80/100)=0.04=4%

That means it is a 96% damage reduction. If an enemy wants to kill you with normal damage, he has to deal 25 times your max hp in order to do so. A 100000 damage will only reduce 4000 hp from you. So, with this figure alone, I don't need to elaborate on how OP he is.
2) Another special trait of Wukong's reduction is that it is not constant and it decreases with its hp percentage. The question here is: How fast does the percentage drop with respect to hp?
The formula of the reduction percentage for Wukong is:

Reduction Percentage = X^3 * 60% (below amel II)
Reduction Percentage = X^(1/3) * 80% (amel II or above)

whereby X is your current hp percentage.
As you can see, there is a huge difference between the ameliorated and non-ameliorated version. Since X is always less than or equal to 1, X^3 has a much smaller value than X^(1/3). To illustrate this, lets say you have 50% hp left, the non-amel version will only have 1/8 of its max reduction left, that is 7.5% while amel version will have about 79% of its max reduction that is 63.5%. See the huge difference? Even at 10% hp left, the amel version will still have 37% reduction percentage. Again, the amelioration makes wukong overcome one of its old weakness, that is multiple attacks.


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3. STRATEGY PLANNING AND EXECUTION

With all the ground works done, here we go to the analytic part.

3.1. STAGE ANALYSIS

In order to clear any particular stage, the first step is always go and look for information. One does not simply fight a battle without knowing his opponent. There is no such thing as 1 team capable to clear every single stages in the game. So, in order to get a strategy layout done, information is very important. To get information, you can either get it from wiki or DSI section of forum. About how to read the table, it is already explained in previous section.

Now that we have the information, we are going to formulate a team based on it.
In TOS, there are generally 3 kinds of team and I am going to cover them seperately.
1) Offensive Team
As the name implies, this team focus on dealing tons of damage to enemy consistently with minimum stalling. Therefore, the leaders used are generally multiplier leader while actives are booster and converter in general with some utility if necessary. The key parameter of an offensive team is its basic damage, so please have a rough idea on how much it is. It can either come from experience or if it is your very first time using the team, please do the calculation using formula from previous section accordingly. About its stalling capability, it is generally its recovery with 1 set of heart rune with your average combo per turn. If your opponent hit somewhere around that, then it should be okay to stall around him, else, you may need to kill him ASAP with skill or without skill.

Here are some general things that an offensive team should look for:

a) Can I defeat this enemy without using any skill?
If yes, congratulation, this particular wave have nothing to concern. Else, please figure out the reason why you cannot defeat it. It can be insanely high defense, lack of damage or annoying skill. Figure out the corresponding active that you need to use to overcome it. Then, figure out whether there are any stages before it that can allow you to get the active ready. Besides that, figure out whether you can reserve the skill until that stage as well as do you need that stage before that round. Planning for offensive team is mainly about planning of usage of skill. Therefore, it is advisable to use a team with members of very high or max skill level.

b) Where can I stall?
As previously mentioned, actives are important for offensive team. It is important to find places for you to slowly grind back the skills you need. In general, places to grind are enemies that cannot kill you, preferable high hp but low attack/high cd since you have high attack as an offensive team. In addition, killing an enemy is considered 1 round, therefore, if your team is running on low cd skills, it might not be necessary to grind at all since you can just get back your skill along the way while killing enemy.

c) Is there any enemy that I have to tank some damage no matter what?
This is very common nowadays, with all sorts of initial shield, ambush shield, tumbler shield, kamikaze, etc. If you have enough hp or recovery to handle this, then it will be fine. Else, you may need to include member with actives that can help you to survive. This is grave for YOLO team like enchantress team or crescendo team as these teams generally cannot take any damage at all, if you cannot overcome the enemy with any actives, then you should give up on using them.

After taking note of all these, run an simulation yourself to see whether the team that you form can works or not. Figure out how many turns you can stall on stages you are supposed to stall, whether your skill can be ready on time, is your burst enough to kill the enemy, can the enemy kill you, etc.

2) Defensive Team
Defensive team are formed based on the fact that as long as your enemy does not kill you, you don't lose and you can take your time to slowly kill your enemy. There are 3 kind of defensive leader in this game: reduction, tumbler and healing. Reduction is the most common one as it reduces damage directly as long as the attribute is correct. Tumbler is generally used against enemy that can hit insanely high that even reduction won't save you since tumbler makes you immortal regardless of enemy's damage. Healing leader is used when healing is difficult or paired with tumbler leader for pseudo immortality. (Enemy without multiple hit, ignore tumbler, and always come in single will never kill you as long as you can heal back)

For defensive team, things that you should look for are:
a) Is there any enemy that can kill you?
Selection of member for a defensive team is generally based on this. If there is, try to figure out any way to overcome it. You can use active to help for survival (Like PR Verdandi), or just burst with multiple actives together to make up for your lack of leader multiplier. However, it is not always that you can do this as MH often includes enemy with phantom, multiple hit and annihilation to stop defensive team. 

b) Time needed to beat the stage.
While keeping yourself immortal, it often takes very long time for defensive team to beat a stage. Therefore, here comes a the third team: Hybrid team.

3) Hybrid Team
This team takes stall burst tactics to its extreme. When it stalls, it is practically immortal and when it burst, it is instant kill. The first generation of hybrid team is Odin-Daji team. After Wukong's amelioration, Hades-Wukong and Wukong-Daji become the most popular team known to clear many stages safely. Team members used are often boosters and sometimes utility to help healing. With one leader with offensive multiplier, we have easier time in killing enemies that won't kill us as our base damage is quite good. When we need to burst, we can use less booster than defensive as we have 1 leader multiplier, therefore, we can burst higher and consecutively.
For Hybrid team, you just need to identify enemies that can kill you and use your burst for it. Otherwise, just kill your enemy normally while stalling for actives.

3.2. STALLING AND BURSTING

Stalling
Stalling is all about how to kill your enemy as slowly as possible while not letting yourself die. In order to do so, you need to have low damage while having enough recovery which is kinda contradictory as the prior need low combo while the latter need high combo. This can be easily done by defensive and hybrid team. It is quick tricky for offensive team since your base damage is too high and to restrict it, you will restrict your recovery. Therefore, it is important to know how to stall.

General procedure is as follows:
i) Kill other enemies and left with the most non-threatening one.
ii) Start dissolving at low combo at a time if recovery is not needed

iii) Kill your enemy when active is ready.

For (ii), it can be troublesome from time to time as you will short of recovery and die from it. Therefore, we need some ways to overcome it.
i) For team with Daji ally, you can dissolve your attack rune as 1 combo. This makes Daji's leader skill useless as she is 1x multiplier at 1 combo. Similar for team with Circe on, you can purposely dissolve all your attack runes but less than 4 combo. By removing attack runes from board, you can dissolve more combo for healing.
ii) If your team is using heart converter with column amelioration, you can place 4 attack runes under its column and use its active, this helps to remove 4 attack runes at once.
iii) Purposely miss the condition for enemy's skill such as enchanted shield and combo shield. This helps you to maintain high combo while not killing your enemy.

3.3. RUNE SPINNING AND STACKING

To be updated.

3.4. DIAMOND USAGE

To be updated

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#3
4. RESOURCES MANAGEMENT

It is insufficient to attempt stages with just strategies and skills. Card collection and training is also part of the game. Therefore, it is important to know how to distribute your resources among your collection so that you have more options opened to you in your strategy planning. It does not matter whether you are F2P or P2P player, it is important to have a priority in your feeding, being a P2P only means that you have more diamonds to spend on stamina refills to get resources.

4.1. LEVEL

Leveling is the most basic way to upgrade a card. After each level up, the card's stat will be improved by a preset amount. You will have more hp, more attack and more recovery which determine the performance of your deck. Therefore, level up is the most basic yet most important aspect in the game.

In TOS, a card's level is determined by its accumulated exp. The value of accumulated exp required for a certain level is determine by the following formula:

Required Accumulated EXP = [(N-1)/98]^2 * X



whereby N is the level and X is the preset value which determine the EXP curve of a card. In general, most end game cards in TOS will have 3 million to 8 million for the value of X. X is also the EXP required for a card to reach level 99. To know the value of X, there are 2 ways to do so. First, you can get the level 1 version of the card and find the exp needed for it to reach level 2, multiply it by 98^2 and you get the value of X. Second, you can just go to TOS Wiki gallery section and look for exp curve of the card to get value of X.

For a card with max level of 99, the 1st quarter of exp will actually bring a card from level 1 to 50 while the last quarter will only bring it from 85 to 99. This explains why it is so difficult to level up a card in its higher level.

Although I said it is difficult to level up at higher level, having deck with all max level cards is pretty much the norm of the game. When a card reaches its max level, it does not necessarily means the end of improving the cards. Currently, TOS have two ways to improve a card beyond its level limit.

1) Evolution
For certain cards, when it reaches its max level, you will have the option to evolve it by gathering the specified cards as sacrifice. All you need is a few clicks and the evolution will be done. However, take note that the accumulated exp will be reset back to 0 and level will be reset to 1 so that you can continue to feed card to it to increase its accumulated exp for leveling. Also take note that after evolving, the stats of the card may drop since the level drops but no worry, the final stats of the card will always be higher than its previous stage. Finally, the exp curve for the evolve card may also change (usually increase) so it is normal to have a evolved card more difficult to level up.

2) Power Release
Besides evolution, TOS also have power release to further improve your card's limit. In order to power release, you also need to gather special materials as sacrifices and usually it is harder to get power release material than evolve material. However, unlike evolution, your card does not have to be max leveled but you have to pass a specific stage with only the power release candidate. If you manage to pass it, then your card will be power released, else, you will waste 30 stamina for a fail attempt and the materials will not disappear. Another difference between power release and evolution is that accumulated exp will not be reset for power release. However, your card's level may drop after power release as power release may change the card's exp curve. There is no difference between power release at high level or low level as your accumulated exp remains the same, but I would recommend power release at higher level so you can pass the stage easier.

Fusion
There is one more feature in this game that affects a card's level and that is fusion.
As the name suggest, fusion is a game mechanism that allows two cards to combine together as a new card. So, what is the level of the card after fusion? This again have to do with accumulated exp.
Lets say card A has accumulated exp of P and exp curve of X while card B has accumulated exp of Q and exp curve of Y, after they fuse, the card will have accumulated exp of P+Q and exp curve of X+Y.
By use this two values and apply to the equation
Level = floor{[(P+Q)/(X+Y)]^(1/2)*98+1}


which is essentially the inverse formula in accumulated exp section.
Lets do a sample calculation:
A golem of max level with 4million exp and exp curve and a melog with 0 exp and 4 million exp curve fuse together, the resultant card have exp curve of 8 million and accumulated exp of 4 million, apply this to the formula above, we get the level = 70 which is very different from taking the mean value.

4.2. SKILL LEVEL

Besides level, another way to improve a card is by skill leveling. Every single card in TOS has an active skill for bursting/control/direct damage/damage reduction/conversion/etc. These skills are indispensable for every single team. Therefore, the faster your skills recharge, the stronger your team is. In order to speed up the recharge, you need to lower down its number of turn for cooldown(CD). The primary method of doing so will be leveling the skill.

Skill level of a card depends on a seperate variable from accumulated exp, which is known as accumulated battle round. This variable refers to the number of stage's round a card has undergo. For example, if you finish a stage with 5 rounds, then the accumulated battle rounds of all cards in the deck you use will be increased by 5. One important note here is that the increment strictly refers to the no. of round of stage and not the number of round you spent to clear the stage, therefore, stalling in a stage does not help you to get more battle round.

Required accumulated battle round for each skill levels is stated below:
Skill LevelRequired Accumulated Battle RoundAdditional Battle Round to Next Skill Level
1016
21619
33550
48575
5160300
6460500
79601800
827602500
952603000
1082603000
11112603000
12142603000
13172603000
14202603000
1523260
 

As you can see, skill leveling through accumulating battle round is extremely tough at skill level beyond 8. Therefore, we also have alternate methods for skill leveling.

1) Feeding Cards of Same Skill
Doing so will grant a chance of skill up. When it happens, the accumulated battle round will be increase to the beginning of next level. For example, your card have 2500 battle rounds (slvl 7) and you get a skill up from this, your new battle round will become 2760 which is beginning of slvl 8. Besides that, you also get the accumulated battle round of the card added to the main card. If it is less than 10, then 10 is added. This method is generally used for farmable cards whereby you can get many copies of a same card. Generally, I suggest doing so after the card is slvl 7/8 so that you gain more battle round in return.

2) FeedingHarpy/Baby Harpy
By feeding harpy to a card, you will get a guarantee skill level up for your card and accumulated battle round will be beginning of the new skill level. If you feed baby harpy, the accumulated battle round will be increased by 200. If we compare these two cards, baby harpy is generally used to get your card to a moderate skill level while harpy is used for maxing the skill level. From the table above, 1 baby harpy will get your card to slvl 5 and 5 will get it to slvl 7 which is pretty good. For harpy, it is optimum to use it from slvl 9 onward since from there onward it will be 3000 rounds per slvl. However, if you are less patient, you can also start feeding at slvl 8 since you only lost 500 rounds from that. It is also advisable to feed harpy right after your card get a skill level up so that you don't waste the extra battle round.

4.3. AMELIORATION

To be updated

4.4. RESOURCES DISTRIBUTION

To be updated

5. MISCELLANEOUS 

For whatever numerical topics that I cannot come up with a specific section to cover.

5.1. SEAL CARD DRAWING

We always hear about ranting for not getting a specific cards during a seal card draw. Lets analyse this mathematically.
First of all, we need to know how many possible cards we can get from diamond seal:
  1. Greek x 5
  2. Norse x 5
  3. Egyptian x 5
  4. JTTW x 5
  5. Egg Elves x 3 x 5 (3 possible stages)
  6. Ethereal Dragons x 2 x 5 (2 possible stages)
  7. Enchantress x 3
  8. Zodiac x 12
  9. Immortal Fiend x 5
  10. Dragon Spiritor x 5
  11. SS Ducks and Dogs x 5
  12. Blood Fiend x 5
  13. Chinese Investiture x 5
  14. Mechanical Scholar x 5
  15. Tribal Beast x 3 x 5 (3 possible stages)
  16. Melog x 5
  17. Immortal Fiend Partner x 5
  18. Youkai x 5
  19. New Dragons x 5
So, in total, there are 125 possible cards right now for you to draw from diamond seals.
Even if MH make it such that each card have even likely chance to get, it will be 0.8% for each card.

So, if you are aiming for a specific card during bonus, what will be the chance for you to get a particular card for every draw?
Given attribute bonus x 2 and additional x 2 for specific card, it will be 3.2%.

Lets say you are making 10 consecutive draws, what will be the chance for you not getting the card you want?
0.968^10 = 0.7224 which is 72.24%. So, it is mathematically normal for you not to get what you want if you are making so little draws. In order for you to have at least 50% chance of getting the card you want, the no. of draw you need is 

log_{0.968}(0.5) = 21.3124...




In order to have at least 50% chance of getting a specific card, you need to make at least 22 draws. MH don't really have to greatly alter the probability to screw you in diamond seal, even if they do it honestly and distribute the chance evenly, the vast number of possible card will screw you. So, don't shout and scream at MH if you are making 10 draws and fail to get what you want. Even if you make 50 draws, there is still 20% chance for you to not getting it.

6. CREDIT

Special thanks for anyone that have make contribution to this guide, feel free to point out anything that you find wrong with.

1) Agnesi
For correcting the dragonic compulsion damage boost when score is more than 5000
2) azure86
For correcting the damage reduction formula for wukong with amel II or above.

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#4
Looks like a general guide to tos instead of a specific guide for F2P players
Previously abra kadabra, but ate some shit and evolved.
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#5
Definitely a guide that only OCD Arigato can pull off Angel . Also, agree with alakazam that this is actually more to ToS players in general, not only F2P's.
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#6
@[email protected]
I'm surprised anyone's bothered to do (or plan to do) so much calculations for TOS...
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#7
(03-08-2015, 03:36 PM)quzf Wrote: @[email protected]
I'm surprised anyone's bothered to do (or plan to do) so much calculations for TOS...

(03-08-2015, 03:33 PM)Crux Wrote: only Arigato can pull off  Angel .

There. Angel (compliments from the bottom of my heart *no sarcasm intended)
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#8
(03-08-2015, 03:36 PM)quzf Wrote: @[email protected]
I'm surprised anyone's bothered to do (or plan to do) so much calculations for TOS...

I always have a calculator besides me when I am attempting difficult stages.
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#9
Looking forward to the completion of the guide!
Although I considered myself P2P player, anything that helps to reduce my usage of diamonds are very much welcomed!
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#10
Quote:If you activates it, just multiply them to the damage accordingly.
Next, the score itself also contributes as damage multiplier. The formula is
1 + Score/10000

I don't know if you saw the DC guide, but we figured out that the damage boost is lowered for scores over 5000.

For scores less than or equal to 5000:

Damage Boost = 1 + Score/10000

For scores over 5000:

Damage Boost = 1.5 + (Score-5000)*0.25/10000
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