Math at Murder School: Understanding the Battle Calculations of Fire Emblem Three Houses

Fire Emblem: Three Houses is perhaps one of the most accessible entries in the series. With the ability to play without permadeath in addition to normal mode being easier than past games, Three Houses offers the opportunity for newcomers to join the battle for the first time. Just like the students in their house, these young (or old) minds are honed through regular instruction in the arts of battle. You’ll learn over time the basic instincts that come with battlefield experience. “Maybe I won’t move to that tile,” you say as seven red aggro lines converge on your party’s priestess. You probably have a basic idea of how damage works, too – your steel sword does more damage than your iron one, and magic is more effective against enemies with a lower resistance stat. Still, there’s a lot going on behind the scenes that drives those mechanisms and, by extension, the tactical decisions you make during combat or during your battle preparations.

Maybe you’re wondering how to be the most effective against special unit types like armored units or cavalry. Or perhaps you’re not sure how detrimental it would be to have your squishy magician carry a shield to buff their fragile defenses. Heck, maybe you’re just trying to figure out what the heck the luck stat does. The purpose of this class – er, guide – is to help you to understand the math going on behind the scenes so you can become the ultimate tactician and professor for your spunky students. We’ll start with defining some of the key terms and then work our way through various calculations relevant to combat while discussing how to use that information strategically on the battlefield.

Three Houses Not My Strong Suit

KEY TERMS: UNDERSTANDING YOUR UNIT’S STATISTICS
Statistics, or “stats” for short, refers to the key numerical aspects of a character. These are the core numbers that the various battle calculations in the game will reference. Characters and equipment both have statistics, and each has some different terminology to remember. We’ll start with characters first and then talk about equipment.

The most important stat for a given character is their Health Points, or HP for short. This indicates how much damage you can take before dying. When HP hits zero, the character will either be out of play until the next battle (casual) or permanently killed (classic).

Next we have Strength (Str) and Magic (Mag). These stats determine your physical and magical attack power, respectively. Strength also influences the character’s ability to carry heavy equipment, while magic also impacts the amount of healing that a character is able to do with a spell.

Speed (Spd) is, in my mind, the most important stat in Fire Emblem. It directly influences a battle statistic called attack speed that decides how many times you attack or get attacked during a combat exchange. Attack speed has both offensive and defensive applications, and it also influences your ability to avoid enemy attacks.

Next we have Dexterity (Dex) and Luck (Lck), which aren’t always used together but often influence one another. Dexterity and Luck each have one key battle calculation they each influence alone, and then some which they influence together as a pair. Dexterity has the largest impact on accuracy with physical attacks, while luck is important for avoiding critical hits from the enemy. Together, dexterity and luck both influence magical accuracy and critical hit rate. It’s also worth noting that these stats are typically the ones referenced by abilities with a random activation rate such as Miracle or Lethality.

Three Houses Walls and Lines
I imagine this was the book Ashe was reading just before he learned Dexterity+4.

To help you accomplish your mission of not dying, Defense (Def) and Resistance (Res) are quite important. These influence your ability to reduce damage from physical and magical attacks, respectively. These can be bolstered by equipable shields and also by your battalions.

Finally, there’s Charm (Cha), the newcomer to Fire Emblem: Three Houses. Charm influences the hitting power of your gambits, the special attacks performed by battalions. It’s also important for unlocking the dancer class for one of your characters during the course of the game.

When it comes to your gear, weapons, staves, and shields all have different statistics to reference. The damage dealt by a weapon or spell is referred to as its might (Mt) while the chance it will actually hit the target is the accuracy (Acc). Weapons also have a critical hit chance (Crit) and a weight (Wt). Range (rng) is also important – it determines how far a weapon can reach, measured in a number of tiles. Shields and staves also have weight, but instead of might or accuracy they instead give a flat increase to one of your other statistics like range or protection.

Most of your core battle calculations are actually going to be the sum of a character stat and a weapon stat. Attack (atk) represents how much damage you are going to do while protection (prt) and resilience (rsl) represent how much damage you’ll subtract from the enemy’s attack. Attack Speed (AS) is how fast you are after factoring equipment weight, and Avoid represents your chance to dodge attacks. You have a Crit rate which influences how likely you are to do increased damage, and a hidden stat called critical avoid (sometimes referred to as dodge) which reduces the likelihood that an enemy critical hit will put you in the ground. So now that we know what the battle statistics are, let’s actually talk about how they’re calculated.

Three Houses Study to Destroy Your Enemies
She seems so cute and wholesome, and then she says things like this.

BATTLE CALCULATIONS: USING STATISTICS TO ANTICIPATE OUTCOMES
Once you know the stats of your units and your weapons, you can start to use them in tandem to determine what will happen in battle between you and your opponent. The easiest calculations in the game are the ones related to damage:

Attack = Weapon/Spell Might + Strength/Magic (depending on whether the attack is physical or magical) + Ability Bonus + Battalion Bonus + Art Bonus + Link Bonus

Protection = Defense + Shield Protection + Ability Bonus + Battalion Bonus + Terrain Bonus

Resilience = Resistance + Shield Resilience + Ability Bonus + Battalion Bonus + Terrain Bonus

Damage = Attacker’s Attack – Defender’s Protection/Resilience (again, depending on whether the attack is physical or magical)

Notice that both abilities and battalions can have an influence here. An example of an ability that would influence the damage formula is something like swordfaire – it gives a flat +5 to your attack stat. Many battalions give bonuses to battle statistics, and protection and resilience can both be positively influenced by the presence of a battalion as well as attack. There are other factors too. When you choose to use a combat art, the extra damage from that art is added directly to the attack value of the attacker. It’s important to note that art bonus is added to attack and not damage – if the enemy’s defense is high enough, they can still totally nullify all of the damage from a combat art. A link bonus happens when two characters with a support relationship fight within a short distance – it can increase your damage depending on the nature of the support. Then there’s terrain, which can reduce the amount of damage taken by the unit using the terrain. Forests and defensive tiles are a great place to position your units to reduce the damage that they take during combat.

Three Houses Reduce Your Miss Rate

All of this assumes that your attack is actually going to make contact with the enemy. The calculations for hit are a little trickier because there’s divergence based on whether the attack is physical or magical:

Physical Hit = Weapon Accuracy + Dexterity + Ability Bonus + Battalion Bonus + Art Bonus + Link Bonus – Bow Range Penalty

Magical Hit = Weapon/Spell Accuracy + ((Dexterity + Luck)/2) + Ability Bonus + Battalion Bonus + Art Bonus + Link Bonus – Bow Range Penalty

Whoa, what the heck is going on with that magical hit rate? It turns out that instead of using just dexterity with magical abilities, there’s a little bit of luck involved. You actually average the two together, round down, and add the result to your hit total. Notice also that while physical hit only references weapons, magical hit can be weapons or spells; magical weapons like the Levin Sword or Bolt Axe use the magic formula instead of the physical formula. That’s also the only time the magic formula can benefit from combat arts – spells aren’t compatible with combat arts, but magic weapons are. Note that when firing a bow and arrow, there is a negative adjustment of a flat 20 points to hit for every tile away you are from the target beyond 2. This is why the archer mastery ability Hit+20 is so valuable – it effectively gives you the third tile of range for free and reduces the penalty for being four or five tiles away to something a bit more manageable.

Hit rate doesn’t just operate one-way, though. The defender throws their avoid into the mix, but to find avoid we also have to calculate the battle statistic that avoid draws from: attack speed. This is where the weight of your weapon and other gear becomes a serious factor.

Attack Speed = Speed – (Equipment Weight – Ability Bonus – (Strength/5))

Avoid = Attack Speed + Ability Bonus + Battalion Bonus + Link Bonus + Terrain Bonus

Percentage Chance that an attack will hit = Attacker’s Hit – Defender’s Avoid

Three Houses Empire Mathematicians
I wonder if they’ll accept my two years of experience in data analysis on an application…

So let’s talk about attack speed a bit more. Equipment weight refers to the combined weight value of both the weapon you have equipped as well as the accessory such as a staff or a shield. Equipment weight is reduced in only two ways. One-fifth of your strength is subtracted from it, and there are a few abilities in the game such as Weight-3 which subtract directly from the weight value. If you’ve seen those abilities and wondered how valuable they are, the answer is very. They essentially could say Speed+# because they reduce the amount that your equipment subtracts from your speed in a much more direct way than strength does. Of course, it’s important to note that if the value in parentheses is a negative number, it will round to zero; you can’t increase your speed by having equipment lighter than your strength.

There’s one other thing to note about attack speed, and that’s the possibility of double attacking. When a pair of units exchange blows in combat, they each get one shot at the other, but one may get to attack two times if they have enough of a speed advantage over the other. Weapons such as gauntlets or the various Brave weapons double the number of attacks you can execute when you are on the offensive – this allows a fast attacker with a brave weapon or gauntlets to hit four times in a single exchange. The speed advantage necessary to double attack is 4. As an example, let’s say we have a myrmidon with 9 AS facing off against a fighter with 5 AS. The myrmidon has an iron sword and the fighter has iron gauntlets. When the myrmidon initiates the attack, she will attack twice while the fighter will only attack once. When the fighter initiates combat, his gauntlets will activate, allowing him to attack twice, but the myrmidon will also get to attack twice because she has 4 more speed.

So now let’s look at some special case damage scenarios. The first one we’ll touch on is the critical hit:

Critical Hit Chance = Weapon Crit + ((Dexterity +Luck)/2) + Ability Bonus + Battalion Bonus + Art Bonus

Critical Avoid = Luck + Ability Bonus

Critical Damage = Damage x3

So your critical hit rate is influences by your dexterity and luck together, but it also benefits from bonuses from your battalion and equipped abilities. Just like the chance to hit the enemy, your crit chance is the percentage likelihood that a critical hit will happen. So a crit of 1 means that you’ve got a 1% chance of getting a crit, while a crit of 50 means you’ve got a 50% chance of getting a crit. When a critical hit does land, it multiplies the final damage value of the attack by 3. So note that if you’re doing 0 damage with your attack, a critical hit will still do zero damage.

Three Houses Apply to Other Problems

Next let’s talk about weapon effectiveness and bonus damage. There are various types of special units that can be on the field: armored, cavalry, and flying are the most common, but there are monster types that come into play as well. Some weapons are effective against a unit type while some combat arts grant bonus damage against a unit type. Let’s see how those are different:

Attack with bonus damage = (Weapon/Spell Might x2) + Strength/Magic + Ability Bonus + Battalion Bonus + Art Bonus + Link Bonus

Attack with effective weapon/spell = (Weapon/Spell Might x3) + Strength/Magic + Ability Bonus + Battalion Bonus + Art Bonus + Link Bonus

If one of these is standing out to you as the superior option, well, that’s because it is. Effective weapons are always going to be better for increasing damage against special targets as opposed to combat arts which deal bonus damage. In fact, the only situation in which this would not be true would be when the effective weapon or spell has a weaker might than the weapon with which you were using the combat art. Theoretically, for example, an effective weapon with only 5 might would be worse than combat art with bonus damage using a weapon with 10 might (5 x 3 = 15 versus 10 x 2 = 20). But in most situations you will get a lot more mileage out of simply using a weapon with the effectiveness that you want. This is particularly true because most combat arts don’t have follow-up attacks, so you’re performing one attack at double might instead of potentially two attacks at triple might. The choice is clear.

Three Houses Know to Command Troops

So then, you know how damage and accuracy work, and you know how messed up you’ll get if someone nails you with an effective blow or critical hit. What the heck do you do then? It’s time to break out the healing magic! White magic spells which heal your party are essential tools for success on the battlefield, so let’s look at the math behind how much they heal:

Healing = Spell Healing + (Mag/3) + Ability Bonus + Staff Bonus

Because your magic stat is divided by 3 when healing, it actually has pretty low influence over your overall healing power at lower levels. The bonus healing from abilities like the Bishop’s White Magic Heal+10 or from an equipable item like the Healing Staff (which says it “boosts effectiveness” but simply gives a flat +10 as well) are going to be way more valuable to you than the character’s magic stat until they reach a pretty high level, and even at high levels those +10s are key to gaining quality benefits from healing.

For now, that’s going to bring us to the end of math class! I want to expand the class a bit to add the calculations for gambit accuracy and damage but at this point I don’t have those worked out just yet. I’ll add them in if I learn them at some point. I hope this article has been helpful for you and helped you to learn a couple of useful strategies. For me, understanding these calculations increased the degree to which I value key abilities like Weight-3, Hit+20, and White Magic Heal+10, and it also changed my strategy to prioritize effective weapons over bonus damage from effective combat arts. If you have any questions after reading this guide, or would like to make a suggestion for another guide here on the blog, please feel free to leave a comment and let me know your thoughts!

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