*In progress*

## Hit Force

Lately I shifted my number crunching activities from Armor Enhancement (see my previous blog posts) to The Epic Boss.

Ever since I started playing K&D I was curious how the force of attack is calculated and how it connects to the numbers we see in the game. How come for instance that sometimes when you fight another knight you need a lot of hits and the other one needs a lot of hits till eventually one dies, and sometimes you need only 1 or 2 hits and the opponent does a lot of damage to you per hit as well. I wanted to find a way to crack the code that leads to the attack strength. Now to do that there is a great initial problem. You do not know the stats of your opponent and to calculate those stats you have to make assumptions about how the attack force is calculated. Since that is the very thing I wanted to figure out I couldn't do a lot about it. I needed to find a way to know the opponents stats. I was kind of stuck there, until I realised that the one place in the game where you *do* know those stats are in the Epic Boss fight. At every level the stats of the EB are given. So it must be possible to figure out how the force of the hits you and the EB give are calculated. And that is just what I did.

## First Observations

DecuGamer wrote a blog about it: Math_in_Knights_and_Dragons_pt.1. He gave a few formulas there that sound plausible and that can (I can say now) act as basic formulas to calculate it all. However, if you just fill them in and use the info in his blog, you will not get accurate results. So although the formulas were a great starting point to me, they needed some serious adjustment.

In my search for that I followed a few dead ends. The first thing I did not accept in the formulas was the concept of the *Magic Number*. So I set out to calculate the Magic Numbers based on the attack/defense stats of the Epic Boss. And I succeeded for Bosses Asherah, Wicker Warrior and Abominable Snowman. But then came Jack Frost. Jack Frost had totally different attack/defense stats, but it turned out that in most levels the Maginc Number was actually the same as for the other Bosses. So I had to accept that

- I can not calculate the Magic Number
- The MN is not 100% consistent from 1 Boss to another. Although most probably the MNs were similar for all (or most) Epic Bosses previous to Jack Frost, we have to accept the fact that it is a hidden stat that may be changed without notice
- item 2 makes that we can use the MN to predict what strength we need against the Epic Boss on any level, but we can not be 100 % sure if we are calculating the right value.

Another important observation I made, is that the force of attack is linearly dependent on the attack stat of the actor and the defense stat of the receiver. In normal human language that means that if your defense stats get twice as high, the damage you receive is halved. And if your attack stat is twice as high the damage you inflict to your opponent doubles, assuming the strength of your opponent does not change.

*Special Attacks* (SA) are *always exactly 1.5 x stronger* than normal attacks. The Epic Boss gets (about) the same number of special attacks as you do. There are sometimes at some random moment hits in the game that have the same force as a special attack. Special attacks create an uncertainty that can never be totally tackled by calulations. You have to accept the fact that you can loose when you are in fact stronger, because the EB hits you more often at SA strength. On the other hand you can have (and on numerous occasions I did have) a lucky shot where an extra SA strength attack or a better timed SA attack gave me the advantage and a win against the odds. It is possible to calculate the odds of winning with a certain amount off overkill, but I am not planning to do that in this blog series. In the same way as specials attacks, missed hits also create uncertainty about the outcome.

## Elements

A fourth observation I made is also important. DecuGamers' formulas did state factors for elemental advantage, but the way they were presented and the values made quick calculations with them tricky. I found they are basically right though. So what I did was just shifting the basic value for the elemental advantage from single strong armor to neutral armor. At first sight the figures make more sense then. DecuGamer said that the factor is 1 for single strong armor, 0.(6) for neutral, and 1.(3) for dual strong armor. My method changes that to 1 for neutral armor, 1.5 for single strong armor and 2 for dual strong armor. The figures work out basically the same but they look a bit easier.

But before I end up in mathematical details and figure wizardry in later posts on this subject, today I will give a few tips for choosing and comparing armor and the way to choose the strongest armor to fight the Boss. Unfortunately there is a tiny bit of maths involved in it. But I will keep it to a minimum.

First of all you have to understand the concept of *Elemental Advantage*. Anyone who plays this game longer than just a few days will have noticed that every armor contains 1 or 2 Elements. There are 6 elements in the game, 5 of which are composed in a circle of elemental advantage.

Every armor in the circle has 1 element that is strong against it, and 1 element it is strong against. That is well known I guess. But how does that *work*? What does "strong" actually *mean*? I found that out in detail. Here is what I found (I already posted the essence of this story on the Elements page):

*Elementally weak* does not exist. Only elementally *strong*. If an armor is strong against another, the damage it does to the other armor is increased, but the damage it receives from the other armor does not change. In return If the other armor is strong the damage it does to your armor increases, but the damage you do to it does not. If you have time to live you do exactly as much damage to it as against an armor with no elemental advantage (elementally neutral) and you need exactly the same number of blows to kill it. That means that the somewhat enigmatic situation can exist (and does exist with Starmetal Armor) that an element can be *strong against itself*. That basically means that the damage done to the other armor increases, and at the same time the damage received from that armor increases too.

### Single Strong - Dual Strong

But how much then? Well, by the elemental advantage factor. If you attack the Epic Boss with a single strong armor, your blow is multiplied by 1.5. That means that your armor with a single elemental advantage is as strong as an elementally neutral armor with stats as much as 1.5 x higher. And that is not only the stats in your armor list. It is the stats of *your knight wearing the armor*. If your armor has an attack of 900, and your knight adds 250 to it, the total attack will be 1150 and be 1.5 x stronger. So it will be worth as much as a neutral armor with a total attack of 1725. That means your elementally strong armor of 900 attack is equal to your neutral armor of 1475. Given the defense stats are equal.

## Comparing Attack and Defense

But defense stats are hardly ever equal. So how do I compare 2 armors if the stats are totally different? The thing to know for that is that the amount of damage done to you is linearly proportional to your defense. Against an oppononent of a given strength, the damage received by each blow decrease by 10% if your armors strength increases by 10%. To be accurate: if your armor gets 10 % stronger the amount of damage is (damage)/110%, is about 91%. If your defense stat doubles the strength of the blow your receive will be halved.

The damage you receive makes you loose health points. So if your knight has 455 health points and you receive a blow of 110, your health will decrease by 110 to 335. You die once your health reaches zero. So you can count your life in the number of blows you can receive. If your defense stat doubles every blow you receive will do half the damage, and you will be able to receive roughly twice as many before you die. That simply means you can hit back twice as many times, and hence do double the damage.

In the same way if your health increases you can receive more hits and hence your strength increases. Double health = (almost) double hits.

So:

your strength = (number of blows you can receive) * (your hit force)

Given your opponents strength we can now say:

your strength = (your opponents input * your total defense * your health) *(your opponents input * your attack) =(your opponents input) * (your total defense) * (your total attack) * (your health)

To compare your different armors you can simply multiply the total attack and the total defense and health points of your knight wearing the armor. The one who has the highest number is the strongest. With the elemental advantage factor added:

knights'strength = (attack * elemental advantage) * (defense / elemental disadvantage) * health

Again the knight with the highest number is the strongest.

*Note: The survival strength of your knight is measured in number of blows it can receive and not in the defense stat as such. Sometimes an increase in defense will not result in survival of more blows. The opponent will just have less overkill. So this way of calculation gives only an indication of strength. I will give a more detailed (but more arithmetical) method to calculate precise strength in a next blog post.*

## Elemental Advantage

OK, but what is the exact Elemental Advantage factor I have to use? That is determined by the elemental advantage we actually have.

As I already stated on the Elements page there are 3 basic types of armor as far as element is concerned.

- Mono Element armor
- Adjacent Element armor
- Non-adjacent Element armor

### Mono Element Armor

Mono Element armor is the easiest type. It has only 1 element and it can not be dual strong. However it *can* be both strong and weak. If for instance you attack Jack Frost (mono air) with an earth/water armor, the elemental advantages work against each other. The knights strength will be :

knights strength = (attack * 1.5) * (defense / 1.5) = attack * defense

So advantage and disadvantage level out and the resulting strength is the same as if no elemental advantage existed.

## Adjacent Element Armor

Adjacent Element Armor is armor with elements that are next to eachother in the Elemental Advantage ring. One of the elements is strong against the other. This armor can be dual strong, and another armor can be dual strong against it. An adjacent armor is single strong against itself. So if you would attack it with armor of the same type, like in mono element armor the Elemental advantage will make that your hit strength multiplies by 1.5 and your opponents hit strength does too. Like for mono element armor the resulting strength is similar to an elementally neutral armor.

So what armor should you choose if the EB is water/fire like Krampus? The water and fire elements are adjacent in the Elemental Advantage ring (water is strong against fire). So your first choice will be air/water.

Here is a table containing other combinations. The order in which the elements are presented in the armor does not matter.

elements | advantage | disadvantage | result |
---|---|---|---|

earth/air | 1.5 | 1 | 1.5 |

air/water | 2 | 1 | 2 |

water/fire | 1.5 | 1.5 | 1 |

fire/spirit | 1 | 2 | 1/2 |

spirit/earth | 1 | 1.5 | 2/3 |

earth/water | 1.5 | 1 | 1.5 |

water/spirit | 1.5 | 1.5 | 1 |

spirit/air | 1.5 | 1.5 | 1 |

air/fire | 1.5 | 1.5 | 1 |

fire/earth | 1 | 1.5 | 2/3 |

mono water | 1.5 | 1 | 1.5 |

mono air | 1.5 | 1 | 1.5 |

mono earth | 1 | 1 | 1 |

mono spirit | 1 | 1.5 | 2/3 |

mono fire | 1 | 1.5 | 2/3 |

(mono) starmetal | 1.5 (?) | 1 | 1.5 (?) |

If I am correct that covers every element combination available in the game. If you want to fight another adjacent element armor you just shift the elements along the Elemental Advantage circle. if for instance you fight air/earth instead of water/fire you shift water to earth, fire to air, spirit to water, earth to fire and air to spirit.

The advantage/disadvantage is symmetric. There are just as many armor elemental combinations you are strong against as there are strong against you.

To compare the strength of different armor types against a water/fire armor you simply multiply attack, defense and health with the resulting factor.

## Non-Adjacent Element Armor

Non-adjacent elements have some peculiarities. They are dual element armors of which neither element is strong against the other. There are only 5 elements in the ring. That means if there is 1 element between 2 elements, there are automatically also 2 elements between them if you follow the circle in the opposite direction. So there is only 1 type of non-adjacent armor, namely armor with 1 element between the 2 elements.

In principle a non-adjacent element armor can be dual strong. But 1 element of the opposing armor is automatically in between it's own elements, so a dual strong non-adjacent armor always has a single elemental disadvantage.

It has been debated what the elements will be of the Epic Boss after Krampus. Looking at its colors it may be an earth/air (adjacent) armor. But it may also be a (non-adjacent) water/earth armor. How are the element factors if it is?

Here is a table:

elements | advantage | disadvantage | result |
---|---|---|---|

fire/earth | 1 | 1.5 | 2/3 |

spirit/air | 2 | 1.5 | 4/3 |

earth/water | 1 | 1 | 1 |

air/fire | 1.5 | 2 | 3/4 |

water/spirit | 1.5 | 1 | 1.5 |

earth/air | 1.5 | 1.5 | 1 |

air/water | 1.5 | 1.5 | 1 |

water/fire | 1 | 1.5 | 1.5 |

fire/spirit | 1.5 | 1.5 | 1 |

spirit/earth | 1.5 | 1 | 1.5 |

mono air | 1.5 | 1.5 | 1 |

mono water | 1 | 1 | 1 |

mono fire | 1 | 1.5 | 2/3 |

mono spirit | 1.5 | 1 | 1.5 |

mono earth | 1 | 1 | 1 |

(mono) starmetal | 1.5 (?) | 1 | 1.5 (?) |

A single strong armor is stronger than a dual strong armor against a non-adjacent element EB, The single elemental disadvantage that is linked to the dual elemental advantage has a bigger impact on the strength of the armor than the 2nd advantage itself.

## Starmetal

Starmetal Armor is mono element. It is single strong against mono element armors. I can not tell how it behaves against dual element armors because I didn't get the opportunity to test it yet. Maybe it has a single advantage, maybe it has a dual advantage.

## Choosing Armor for the Future

Overlooking it all I can conclude 1 important thing. Adjacent Element Armor can be dual strong and take advantage of that, but non-adjacent armor in practice can not. The same goes for mono element armor. That means that for fighting a mono element or non-adjacent element Epic Boss there is no need to use mono element or non-adjacent element armors against it. If you have them available in the right elemental combination adjacent armor will do just as good. But If you want to fight an adjacent element Epic Boss using the right adjacent Armor gives you a lot of extra strength. So it figures that if you want to save your efforts and want to get as strong as you can against the Epic Boss, it makes sense to work only for adjacent Armor. I did not realize that myself before I started writing about Elemental Advantage. But it made me decide to suspend enhancement of all my armors that are not adjacent element. Instead I throw everything in adjacent armor now.

In the arena you don't know what you will encounter. Dual elemental advantage may be of less value because it works 2 ways: the opponent may have dual element advantage against you as well. Mono element armor will probably perform more constant in the arena, but on average there will be no difference. If you are weaker than average dual elemental advantage will provide an extra chance of winning against stronger players, that is offset by a smaller chance of loosing against weaker players. If you are strong in the arena you won't need the extra win chance of dual elemental advantage that bad anymore, and the chances you encounter a weaker player with elemental advantage over you are bigger. So you may want to choose for the more constant performance of other mono element or non adjacent element armor.

So if I have a maxed out Witch's Robes + (air/earth) and I get Aegis of the Dragon + (earth/air), do I want the (probably) air/earth armor of the next epic boss (after Krampus)? Yes I do.