Hypothetical and Ridiculous HON Math, Your Questions Answered! - For Entertainment Only

[[S2]MacroHard]

Rules

• Simply ask your crazy hypothetical question on this thread!
• Topic can be anything HON related, but it must involve some sort of number crunching.
• If I like your question (and have the capability of answering it), I will post a response on the front page.
• If I do not like your question or am unable to answer it I will respond with a reason why.
• If I straight up ignore your question it is because you should be able to answer it yourself using the resources on the mechanics forum.

Disclaimer

• Everything here is for entertainment purposes only. These answers, while as accurate and truthful as possible, have little (if any) application in game.

Q: Assuming an enemy is running away from you, how fast must the enemy and Witchslayer be such that Miniaturize will be quicker to disable than Graveyard?

Witchslayer must have 375 or more ms!

Miniaturize
0 ms cast action time
500 cast range
0 travel time (instant)

Graveyard
300 ms cast action time
800 range (1600 projectile speed over 500 ms)
500 ms travel time

Variables:
D = distance between self and target
Mw = movespeed of witchslayer
Mt = movespeed of target

Time to Disable with Miniaturize = (D - 500) / (Mw - Mt)
This is straightforward. It's simply how long does it take to get within 500 distance of target, since cast time is instant and there is no projectile.

Time to Disable with Graveyard = {D - [800 - Mt*(0.3 + 0.5)]}/(Mw - Mt) + (0.3 + 0.8)
The [800 - Mt*(0.3 + 0.5)] in the numerator is the maximum cast range to still disable, as the max range is 800 and the delay is the cast time + travel time.
The 0.3 + 0.8 is added again because once you get within maximum cast range, the projectile still has to travel.

If you set these equations equal to each other, everything drops out except Mw!

For Miniaturize to be faster than Graveyard...
D - 500 < D - 800 + 0.8Mt + 0.8Mw - 0.8Mt
300 < 0.8Mw
Mw > 375

What this means is that if Witchslayer has an MS faster than 375 then using Miniaturize is the fastest way to disable an enemy that is running away, regardless of enemy movespeed!

Q: Is it possible to tank the fountain forever with only auras (no active spells)?

Yes!

Martyr
BehemothsHeart / SacrificialStone / Wingbow / FrostfieldPlate / FrostfieldPlate / FrostfieldPlate
(the sac stone is with 99 charges)

Allied assistance:
Empath: +0.2% +3 hp/sec regen
AbyssalSkull: +5 armor
SolsBulwark: +5 armor
Astrolabe: +3 hp/sec
Refreshing Ornament: +2 hp/sec
Vagabond Leader: +3 hp/sec regen

All of that should give you 5750 health, 68.58 armor, 30% evasion, 50% damage reduction (at very little health), and 101.4 hp/sec regen.
The fountain would do (194.5/0.15)/(1+.06*68.58)*(1-.3)*(1-.5) = 88.73 damage/sec

This should last forever, just barely hanging on at near death, with only an Empath and Vagabond Leader standing by for auras.

Anyway I'm testing it right now with ~800% gamespeed. Martyr's HP fluctuates around ~ 300-900 HP at around 10 minutes in, which is possible given Wingbow's evasion being chance-based.

Martyr passed the 1 hour test

Q: When does activating Elder Parasite add more damage than extra damage received?

• D1 = your damage
• S1 = your attack speed (unactivated)
• A1 = your armor
• B1 = your base attack time
• D2 = enemy damage
• S2 = enemy attack speed
• A2 = enemy armor
• B2 = enemy base attack time
• F1 = your damage factor = 1/(1+.06*A2)
• F2 = enemy damage factor = 1/(1+.06*A1)

Extra Damage Output:

damage with +100 attack speed - damage without +100 attack speed
D1*F1*(2+S1/100)/B1 - D1*F1*(1+S1/100)/B1

Extra Damage Received:

(amplified damage - lifesteal with +100 speed) - (regular damage - lifesteal without +100 speed)
[1.15*D2*F2*(1+S2)/B2 - 0.17*D1*F1*(2+S1)/B1] - [D2*F2*(1+S2)/B2 - 0.17*D1*F1*(1+A1)/B1]

Extra Damage Output > Extra Damage Received:

D1*F1*(2+S1/100)/B1 - D1*F1*(1+S1/100)/B1 > [1.15*D2*F2*(1+S2/100)/B2 - 0.17*D1*F1*(2+S1/100)/B1] - [D2*F2*(1+S2/100)/B2 - 0.17*D1*F1*(1+S1/100)/B1]

1.17*D1*F1*(2+S1/100)/B1 - 1.17*D1*F1*(1+S1/100)/B1 > 0.15*D2*F2*(1+S2/100)/B2

D1*F1/B1 > 0.1282*D2*F2/B2*(1+S2/100)

When this is true, activating Parasite will do more good than harm. This can also be applied to multiple targets of multiple damage types...

D1*F1/B1 > 0.1282*D2*F2/B2*(1+S2/100) (values for enemy 1) + 0.1282*D3*F3/B3*(1+S3/100) (values for enemy 2) + 0.1282*D4*F4/B4*(1+S4/100) (values for enemy 3) + ...

Against a single target, or even multiple targets, this will almost always be true (as long as you do not get stunned) such that you can continue your attack uninterrupted. Because of this Shrunken Head should probably be strongly considered while using EP.

Example:

Equal base attack times of 1.7 -> B1 = B2 = B3
Your damage is 180 -> D1 = 180
Enemy damage is 210 and 150 -> D2 = 210, D3 = 150
Your armor is 20 -> F2 = F3 = .455
Enemy armor is 15 and 22 -> F1 = .526 (based off the target you are attacking)
Enemy attack speed is 100 and 85 -> S2 = 100, S3 = 85

180*.562 > .1282*210*.455*(1+100/100) + .1282*150*.455*(1+85/100)
101.16 > 24.50 + 16.19
101.16 > 40.69 ?
True.
Elder adds more damage than damage received, even against two enemies. And as you can see, there is still much room to spare.

Note that your own attack speed drops out of the equation. So elder parasite, when it comes to adding damage compared to damage received, is equally effective whether you have +30 speed or +300 speed. oO

Q: At what combination of attack damage and speed will Duck Boots provide more DPS than Doombringer?

Assumptions

• Primary attribute is Agility

Method

• DPS from Doombringer = (250+D)(100+S)/100-D(100+S)/100
• DPS from Duck Boots = (3+D)(103+S)/100-D(100+S)/100
• Equate these equations and solve for S in terms of D
  
  • D = attack damage before item purchase
  • S = attack speed before item purchase

Answer

• If you have 17,000 damage but only 100 attack speed, Duck Boots will provide more DPS than Doombringer.

Graph

Q: If Valkyrie were at the top of the CN Tower, how long would it take for her Javelin to reach the ground? How far could she throw it?

Assumptions and Parameters:

• Valkyrie is 200 units tall in Newerth, 1.7m tall on Earth
• Valkyrie javelin travels at 857.14 units/second = Vi
• CN Tower is 553m high = h

Solution:

• The conversion rate, based on the height assumptions for Valkyrie, is .0085 meters per unit.
• Valkyrie's javelin therefore travels at 7.29 m/s.
  
  • Vxi = 7.29cos(θ)
  • Vyi = 7.29sin(θ)
  
  
• Per the kinematic equations for a uniformly accelerating body, h = Vyi*t + 0.5*a*t²
  
  • 4.9t²-7.29sin(θ)t-553 = 0
  • t = [7.29sin(θ)+sqrt(53.1sin²(θ)+10800)]/9.8
  
  
• Total distance d(t) = Vxi*t
  
  • d = 7.29cos(θ)*[7.29sin(θ)+sqrt(53.1sin²(θ)+10800)]/9.8
  
  
• Maximum distance achieved when d(d)/d(θ) = 0
  
  • d(d)/d(θ) = 0.744cos(θ*[7.29cos(θ)+53.1sin(θ)cos(θ)/sqrt(53.1sin²(θ)+10800)]-0.744sin(θ)*[sqrt(53.1sin²(θ)+10800)+7.29sin(θ)] = 0
  • θ = 3.996°

Answer:

• It would take 9.906 seconds for Valkyrie's javelin to reach the ground, assuming she threw it downwards.
• If she threw it horizontally, it would take 10.62 seconds to reach the ground, and would land 77.45 meters (254.1 feet) away from the base of the tower.
• If she threw it at the optimum angle of 3.996°, the javelin would land 77.63 meters (254.7 feet) away after 10.68 seconds

Q: How many attacks will it take for Predator to kill a target?

Assumptions

• Level 4 Carnivorous
• Predator's attack damage is only 1% of total enemy HP (3000 HP -> 30 damage)
• Enemy has zero regeneration

Method

• HP remaining after attack = X - .07X/(1+.06A) - 1/(1+.06A)
  
  • X = current HP
  • A = armor

Answer

• A=0, 29 attacks
• A=10, 47 attacks
• A=20, 65 attacks
• A=30, 83 attacks
• A=40, 100 attacks

Graph

Q: How many unique 5-man all melee teams can be made?

Assumptions

• Currently 46 melee heroes

Method

• n = number of elements in the set (46)
• k = desired subset of distinct elements (5)
• Total Combinations = n!/[k!(n-k)!] = (46 x 45 x 44 x 43 x 42) / (5 x 4 x 3 x 2 x 1)

Answer

• 1370754 unique 5-man all melee teams
• This is only 2.79% of the 49177128 total unique 5-man teams possible (with the 92 current heroes)

Q: How much damage can Level 3 Puzzlebox Minions do if they are allowed to attack constantly for their entire duration?

Assumptions

• Level 3 Puzzlebox
• Mauler attack damage = 61
• Mauler attack speed = 1
• Mauler mana burn = 75, 60% as physical damage
• Wizard attack damage = 91
• Wizard attack speed = 1
• Wizard Mana Burn spell = 225 magic damage
• Wizard Mana Burn spell is used at t=0
• Duration = 35 seconds

Method

• Total physical damage per second = (61 + 75*(0.6) + 91)/(1+.06*Ap)
• Initial magic damage = 225/(1+.06*Am)

Answer

• 3303 damage against a target with 20 physical armor and 5.5 magic armor

Table

Q: How do you maximize damage out of your 603 starting gold for a ranged hero?

Assumptions

• Agility is primary attribute

Method

• Total damage from = (X+7)*(1.12)
• Total damage from = (X+12)
  
  • X = starting damage (no items)
  
  
• (X+7)*(1.12) = X+12

Answer

• If your starting damage is 34.67 or higher, will give the most damage for a ranged hero
  
  • Soulstealer's starting damage is 38, so this applies to all ranged heroes
  
  
• You will have 25 gold remaining and two slots open; after only one creep kill you can bring or on the courier!

Q: At what Intelligence should a Vindicator with Harkon's use his Master's Incantation instead?

Assumptions

• Vindicator's average base damage is 29
• Vindicator's only item damage comes from Harkon's (+27 damage)
• All other damage gain is from Intelligence (passive, stats, +int items, etc)

Method

• Harkon's Blade damage = (B+I+D)/[1+.06(Am-5)] = (56+I)/(.7+.06Am)
• Master's Incantation damage = (B+I+D)/(1+.06Ap)+.6I = (56+1.6I+.036IAp)/(1+.06Ap)
  
  • B = Base Damage = 29
  • I = Intelligence
  • D = Item Damage = 27
  • Ap = Physical Armor
  • Am = Magical Armor
  
  
• Solve for I in terms of Ap and Am
  
  • I = (52.08+3.36Ap-3.36Am)/(.12-.0348Ap+.096Am+.00216ApAm)

Answer

• 531.7 Intelligence (when the enemy has 20 physical armor and 5.5 magic armor)
• 94.9 Intelligence (when the enemy has 20 physical armor and 10.5 magic armor)
• 42.5 Intelligence (when the enemy has 20 physical armor and 15.5 magic armor)

Graph

Q: How much damage does Blacksmith Flaming Hammer do?

Assumptions

• 40 damage per second over 7.5 seconds, Magic Armor -4 to 0 over duration
• 0.05 seconds per frame

Method 1

• Magic Armor = (4x/7.5-4), where X = time
• Damage Factor = 1/{1+.06[A+(4x/7.5-4)]}
• Set up integral:
  
  
  • = 1250*ln(0.06A+0.032x+0.76)
  
  
• Integrate from x=0 to x=7.5

Answer (Method 1)

• 248.8 damage (5.5 magic armor)

Method 2

• Set up table with 150 damage instances at 0.05s intervals
  
  • Base damage = 2 per 0.05 seconds
  
  
• Multiplying incremental damage by the magic armor for that instance

Answer (Method 2)

• 248.8 damage (5.5 magic armor)

Table (Method 2)

Q: If you last hit and deny every creep, how much experience will you and you enemy have 1v1 after 10 minutes?

Norroar's excellent thread dedicated to this question has obsoleted my assessment:
http://forums.heroesofnewerth.com/sh...d.php?t=346304

Q: Damage from Amun Ra's Codex5 with Ult, Spellshards, Hellflower, and Nature's Wrath against Pestilence with Flight and EP?

Assumptions

• Pestilence has 5.5 magic armor.

Method

• Codex5: 800 damage
• Spellshards3: 0 armor (no reduction)
• Ra Ult: +30% damage
• Flight: +10% damage
• Nature's Wrath: +28% damage
• Elder Parasite: +20% damage
• Hellflower: +20% damage
• Total Amp Damage: +108%

Answer

• 1664 Damage

Q: How much HP is needed such that holding Behemoth's Heart gives more health than dropping it prior to using bottle?

Assumptions

• Behemoth's Heart adds 965 HP
• Behemoth's Heart gives 0.75% HP/sec regen
• Bottle regens by 135 over 3 seconds, 405 over 9 seconds
• No other regen sources are considered (sorry)

Method

• Regeneration from holding Behemoth's Heart = Regeneration from dropping Behemoth's Heart
• X + 405 + 0.0075*9*H = [(H-965)X/H+405]*H/(H-965)
  
  • X = current health
  • H = max health
  
  
• X(H-965)+405(H-965)+.0675H(H-965) = X(H-965)+405H
• .0675H^2-65.1375H-390825 = 0

Answer

• 2936.6 max HP

Q: At what attack speed will Gladiator achieve perfect synchronization with Flagellation?

Assumptions

• Level 4 Flagellation
• Gladiator attacks constantly
• Gladiator base attack cooldown is 0.59 seconds

Method

• Attack/Second = 0.59*(Attack Speed)/(100)
• Seconds/Attack must divide evenly into the 80 frame window of 4 seconds

Answer

Note: Level 7 Gladiator with no items has 22.2 agility (122.2 attack speed)

Q: At what point will Night Hound do more damage to Armadon attacking from his back than from the front?

Assumptions

• Level 4 Armordillo
• Level 4 Backstab
• No proc items

Method

• Damage to Front: D/(1+.06*Armor)
• Damage to Back: 0.6*(D+A)/(1+.06*Armor)
• 0.6*(D+A)/(1+.06*Armor) > D/(1+.06*Armor)
• 0.6*(D+A) > D
• 0.6D + 0.6A > D
• 0.6A > 0.4D
• A > 0.667D

Answer

• If more than two-thirds (66.67%) of Nighthounds total damage is comprised of agility he will do more damage to Armadon's back

Q: How many level 3 Puppets would have to be chained to one-shot 60 minute Kongor with 1 damage?

Assumptions

• Each Puppet is cast on the one before to make a chain
• No splash damage
• Kongor's buff is level 120 (60 minutes)
• Puppetmaster Ult lt is level 3
• Outer most puppet receives 1 damage

Method

• Each Puppet has 1000 health and transfers 180% damage. Overkill damage is not amplified when transferred to the next target. Because of this, at some point each additional Puppet will only increases the magic damage by 800 damage (180%x1000).
• Kongor has 13500 health and 75% magic resistance at 60 minutes. He can sustain 54000 magic damage.

Puppet n: takes 1 damage
Puppet n-1: takes 1*1.8 = 1.8 damage
Puppet n-2: takes 1.8*1.8 = 3.24 damage
Puppet n-3: takes 3.24*1.8 = 5.832 damage
Puppet n-4: takes 5.832*1.8 = 10.4976 damage
Puppet n-5: takes 10.4976*1.8 = 18.8957 damage
...
Puppet n-11: takes 357.0467*1.8 = 642.6841 damage
Puppet n-12: takes 642.6841*1.8 = 1156.8314 damage
Puppet n-13: takes 1800+(1156.8314-1000) = 1956.8314 damage
Puppet n-14: takes 1800+(1956.8314-1000) = 2756.8314 damage
...
Puppet n-m: takes 800m-8443.1686 damage
...
Puppet 2: takes > 52400 damage
Puppet 1: takes > 53200 damage
Kongor: takes > 54000 damage

When n-m = 0, 800m-8443.1686 > 54000
n > 78.05

Answer

• You would need a minimum of 79 puppets.

Q: What items does a level 25 Doctor Repulsor need to ult indefinitely?

Assumptions

• No regeneration rune

Method
First we have to look at the requirements.

For level 1 ultimate...

• Travels 1250 units per second
• Costs 10 + 1% max mana per 100 units
• R = 125 + .125*M
  
  • R = Required Regen per Second
  • M = Maximum Mana

Putting this in a table...


You can see how the ratio of Max Mana to Regen increases with mana because the fixed values becomes less significant. However, this number (~4-6) is VERY low. Every point of Intelligence gives 13 mana and 0.04 regen, a 325 to 1 ratio. Therefore the more Int you have the more difficult it is to maintain "infinite Ludicrous Speed".

The goal now is to maximize mana regen but keep int (and mana) to a minimum. The only item that begins to compete with this required 6 to 1 ratio is a charged up Sac Stone, which provides a maximum of 99 regen with 400 mana. The ratio for this item is about 4 to 1.

More Sac Stones = better
ANY OTHER ITEM = worse

I set up a simple table to show various combinations of levels, attribute levels, and number of sac stones:


Answer

• A level 6 Doctor can maintain infinite Ludicrous Speed with 4 Sac Stones.
• A level 25 Doctor can maintain infinite Ludicrous Speed with 5 Sac Stones IF he does not level stats nor Ultimate Rank 2.
• A level 25 Doctor can maintain infinite Ludicrous Speed with 6 Sac Stones IF he does not level Ultimate Rank 2.
• A Doctor cannot maintain infinite Ludicrous Speed if he gets Ultimate Rank 2.

Q: How much health/magic armor is required to survive a full combo from a Soul Reaper with SotM, Codex5, Resto Stone, Harkons, Helflower, and Spellshards3?

Assumptions

• Level 25 SR with these items does 381 damage
• Damage from Aura not included
• Health Regeneration not included
• Action order as follows:
  
  1. Hellflower
  2. Single Attack (Harkons activated)
  3. Codex
  4. Ultimate
  5. Refresher
  6. Codex
  7. Ultimate

Method

H = Max Health
A = Magic Armor after Reductions (Magic Armor -11, cannot be less than 0)
X = Damage Factor = 1/(1+.06*A)

• For Magic Armor = 10.5, A = 0.0, X = 1.00000
• For Magic Armor = 15.5, A = 4.5, X = 0.78740

H - 381*1.2*X (attack)
- 800*1.2*X (codex)
- (381*1.2*X + 800*1.2*X)*1.2*1.2*X (ultimate)
- 800*1.2*X (codex)
- [381*1.2*X + 800*1.2*X + (381*1.2*X + 800*1.2*X)*1.2*1.2*X + 800*1.2*X]*1.2*1.2*X (ultimate)
= 0

H = 457.2X + 960X + 2040.8X^2 + 960X + 3423.2X^2 + 2938.7X^3
H = 2938.7X^3 + 5463.9X^2 + 2377.2X

Answer

• For Magic Armor = 10.5, H = 10780
• For Magic Armor = 15.5, H = 6694.1

Q: What is the maximum Crit possible without using Doombringer? With Doombringer?

Level 25 Behemoth no items: 136 max base damage

Riftshards4: +75
Behemoth Heart: +35 str
Behemoth Heart: +35 str
Behemoth Heart: +35 str
Behemoth Heart: +35 str
Behemoth Heart: +35 str
Subtotal: 311 base dmg, +75 dmg
This combination of buffs provides a total of +461% base damage. This means that each Behemoth Heart adds 196 damage... more than Shroud and Insanitarius.

Enrage: +200% base damage = +622 bonus damage

Andromeda: +36% base damage = +111.96 bonus damage
Bloodhunter: +80% base damage = +248.8 bonus damage
Empath: +80 bonus damage
Demented Shaman: +60 bonus damage

Wolf Commander: +30% base damage = +93.3 bonus damage
Abyssal Skull: +15% base damage = +46.65 bonus damage

Double Damage Rune: +100% base damage = +311 bonus damage

:Riftshards: :BehemothsHeart: :BehemothsHeart: :BehemothsHeart: :BehemothsHeart: :BehemothsHeart:
Total Damage: 1959.71
Crit: 4703!


Level 25 Pebbles no items: 279 max base damage

Riftshards4: +75
Doombringer: +250 dmg
Doombringer: +250 dmg
Doombringer: +250 dmg
Doombringer: +250 dmg
Doombringer: +250 dmg
Subtotal: 279 base dmg, +1325 dmg

Andromeda: +36% base damage = +100.44 bonus damage
Bloodhunter: +80% base damage = +223.2 bonus damage
Empath: +80 bonus damage
Demented Shaman: +60 bonus damage

Wolf Commander: +30% base damage = +83.7 bonus damage
Abyssal Skull: +15% base damage = +41.85 bonus damage

Double Damage Rune: +100% base damage = +279 bonus damage

:Riftshards: oombringer: oombringer: oombringer: oombringer: oombringer:
Total Damage: 2472.19
Crit: 5933!

Q: What is the maximum achievable attack speed?

MOA and Aluna can both reach 10 attacks per second, hypothetically.


Master of Arms

Level 25 Agility: 104.4
Elder Parasite: +75
Thunderclaw: +70
Thunderclaw: +70
Thunderclaw: +70
Thunderclaw: +70
Thunderclaw: +70
Self Buff: +500 (repeater + ult)
Empath: +80
Martyr: +60
Predator: +60
Tundra: +60
Daemonic: +15
Puzzlebox: +9
Minotaur: +15
Sub-Total Attack Speed: 1428.4
50% Terrifying Charge Bonus: 714.2
Total Attack Speed: 2142.6
Base attack speed: .59
Unrestricted attacks per second: 12.6413
Unrestricted frames per attack: 1.5821
Restricted frames per attack: 2
Actual attack speed: 10 per second


Aluna

Level 25 Agility: 81.2
Elder Parasite: +75
Thunderclaw: +70
Thunderclaw: +70
Thunderclaw: +70
Thunderclaw: +70
Thunderclaw: +70
Self Buff: +92 (emerald lightning)
Master of Arms: +100
Empath: +80
Martyr: +60
Predator: +60
Daemonic: +15
Puzzlebox: +9
Minotaur: +15
Sub-Total Attack Speed: 1037.2
50% Terrifying Charge Bonus: 518.6
Total Attack Speed: 1555.8
Base attack speed: .69
Unrestricted attacks per second: 10.7350
Unrestricted frames per attack: 1.8631
Restricted frames per attack: 2
Actual attack speed: 10 per second

Q: When does activated Charged Hammer give more damage return than Barbed Armor?

Assumptions:

• Attacking enemies do same damage, D
• Attacking enemies have same attack speed
• Attacking enemies are not using Shieldbreaker nor Harkons
• The lightning proc from Charged Hammer always hits attacking enemy

CH = Charged Hammer
BA = Barbed Armor
D = enemy attack damage
A = Physical Armor
M = Magical Armor
X = probability factor from CH

• X = 0.2 for one target attacking
• X = 0.36 for two targets attacking
• X = 0.488 for three targets attacking

Return Damage from CH = 200X/(1+.06M)
Return Damage from BA = 0.8D/(1+.06A)/(1+.06M)

200X/(1+.06M) > 0.8D/(1+.06A)/(1+.06M)
200X > 0.8D/(1+.06A)
200X+12XA > 0.8D
250X+15XA > D


So if you have 20 armor, Charged Hammer will return more than Barbed armor when two enemies are attacking you with 198 or less damage.

Q: What is the maximum possible EHP (without allied help?)

Level 25 Wildsoul with :BehemothsHeart: :BehemothsHeart: :BehemothsHeart: :FrostfieldPlate: :FrostfieldPlate: :FrostfieldPlate:
Booboo with aemonicBreastplate :AbyssalSkull: :ShieldOfTheFive:
5698 Health, 86.12 Armor, 35139 EHP, 48.74 Regen
212 Damage, 1.538 Attacks/Second
An enemy dealing 350 damage at 2 attacks/second will take 88.0 seconds to kill.

If you want to sacrifice maximum EHP for something almost as good but with more utility...

Level 25 Wildsoul with :BehemothsHeart: :FrostfieldPlate: :SymbolOfRage: :FrostwolfsSkull: :FrostwolfsSkull: :FrostwolfsSkull:
Booboo with aemonicBreastplate :AbyssalSkull: :ShieldOfTheFive:
6553 Health, 67.42 Armor, 33059.5 EHP, 56.51 Regen
314 Damage, 1.818 Attacks/Second
An enemy dealing 350 damage at 2 attacks/second will take 79.7 seconds to kill.

Note 1: A Devourer would need 144 cadaver armor strength to reach equivalent EHP
Note 2: Who needs movement speed when you cannot be killed?

Q: When is +10 attack speed more DPS than +10 damage (for example, toggling Steamboots)?

+10 Damage -> (D+10)*((100+S)/100)
+10 Speed -> (D)*((100+S+10)/100)

For speed to be more DPS than damage,
(D)*((100+S+10)/100) > (D+10)*((100+S)/100)
...
D > 100+S

So if your damage is at least 100 more than your speed, +10 attack speed is more damage per second than +10 damage.

For example, you are a strength hero with 60 Agi (+60) and Steamboots (+30). If your total damage is 190 or more, you do more damage in Agi mode than in Str mode.