"Don't attack skirmishers with crossbows from downhill" Shit, he's not wrong.

"that's a hill i'm prepared to die on" nice one

"What did you do with your math doctorat?" "Over-analyzed a real time strategy game."

*Insert Obi-Wan Kenobi meme here*

7:03 "Even if you won't be pulling out your calculator mid fight.." Bold of you to assume that

People : Math is boring Spirit of the Law: Hold my AOE2 Graphs please

Now I pose another question pertaining to this hill conundrum: Why not take the heat and march your forces up the hill to fight on equal ground? Yes, you're taking free hits, but would that equalize with removing the hill bonus?

The beautiful old intro is a gift from the gods.

"Don't fight with Crossbows against Skirms uphill" *laughs in larry*

"You have two options: Number 1: fight tartars elite skirmishers with crossbowers from downhill, or number 2:" "Its enough! I choose Number 2"

IT'S OVER ANAKIN, MY MEN HAVE THE HIGH GROUND!

So that's the reason to take a major in Mathematics: Age of Empires videos

Anakin tried and it didn't end well.

The intro is back again! :D

1:04 I am so used to a green colour indicating buffs and red colour indicating debuffs that I have been staring at the screen for a good ten seconds in confusion, oops.

man just recently I was thinking those Spirit of the law videos are not as math heavy as they used to be. Really cool to see an in-depth one again! Great job!

Glorious return of the intro song!

Me: It's over [name]! I have high ground! [Name]: oh sh...

7:08 Sometimes I wonder whether or not SotL do research for topic-related jokes

I was watching the playlist, he was casually comparing a few unique units and all was good and all. I fell asleep for a few minutes, when I opened my eyes there were hecking equations and symbols on my screen. I thought I was dreaming and freaked out. It was an otherworldly experience

Obi-wan: "It's over Anakin! I have the high ground!" A: "You underestimate my civ bonus!"

Congrats for playing with Hera. I know you have interviewed Hera and Tatoh in different ocassions. It would be cool to have more interviews with Pros now and then. Thanks SoTL

falls asleep during video sotl: "assuming that you fell asleep" me: did he just read my mind

nice to see some graphs in a SoTL video it feels like it has been a while :)

I appreciate your well written content. I've been following you since your very early videos and I'm always impressed with the quality of your scripts. Thanks for so much great information Mr Law

Ah, I've that opening,thanks for bringing it back :)

Thanks a lot for making this content that I asked on your last video. My approach was like this - Uphill unit size = u Downhill unit size = d Since all the units have equal attack and HP if the damage taken ( or HP loses) over time is same for both group, they will eliminate each other. So u/0.75d = d/1.25u --> d/u = root (5/3) = 1.291 or 29.1% more units. Honestly I'm really surprised to see that the theory applies so perfectly here in practice .

The intro music

I'm currently on my 2nd year on mechanical engineering and seeing the dreaded differential equations being used on my favorite game is a unique kind of motivation

When uphill crossbowman sees downhill skirmisher closing in to attack: :my time has come:

OMG THE INTRO!!

A terrible thought has crossed my mind. One day there will be "Thanks for watching, guys and I'll see you next time." But there will be no next time. Hope this never happens!

I haven't played this game in months, but I still watch these videos as they come out.

People are going to love you for bringing your old intro back

You manage to make your videos about applied mathematics in games both enjoyable and informative at the same time, every time! More videos with numbers and graphs please!!

SOTL: "You should never fight with equal numbers uphill" Liereyy: "I'm gonna pretend I didn't see that"

2000 - Learning history with Age of Empires 2020 - Learning math with Age of Empires Glad that my favorite game of all time never dies.

0:40 Ultra Instinct Villager wears orange clothes... Coincidence? I don't think so

The fact that you calculated an army size of 13 v 10 would equalize the advantage, and the simulated result matched that perfect, is why I FREAKIN LOVE YOU CHANNEL! Thanks for consistently being a boss

It's over Anakin !

God I love that intro so much. Never change

The way this guy explains stuff and still adds in humour is amazing.

How can you still find relevant subjects that help us improve at the game after so long... it amazes me.

Man the memes in your videos are on fire lately. Thanos balance really cracked me up :D

Being a math major student myself, I just love those graphs, formulae, explanations etc.

"Hill bonuses are no joke" Queue Riut's hill castle against MbL in megarandom

Sotl: Damage downhill can be calculated from Lanchester's square law. Me (with 30 champions): TENNOHEIKA BANZAI!!!

"Anakin, I have the high ground.. you can't win!"

You had the perfect chance to use the meme

3:21 PERFECTLY BALANCED AS ALL THING SHOULD BE

I didn’t fall asleep at the first graph but I was picking my brain up off the floor so I appreciate the summary at the end lol

instead of "x - 10" it should be A - B, which leads to A^2 - 2AB + B^2 = A^2 - (b/a)B^2 => A = ((b/a + 1)/2)B So that's basically "cut their power advantage in half". In this case the power advantage is 66%, so the required numbers advantage is 33%.

That was really interesting to see the numbers behind that question...

"Don't attack smirmishers with crossbows from downhill" Unless you're name is Hera

Good news, your doctorate in age of empires 2 has been approved, congratulations!

The best intro

Great video Spirit! You should definitely be invited to a symposium about "Learning Maths through videogames" if there ever was one. Idea on which I would love a dig in. How does ELO rating work in AoE2 DE? Just played a guy yesterday with 2 wins and 9 losses that had a 1345 1v1 elo... How is that even possible with such a limited amount of games?

Math: It's a beautiful thing.

7:10 the brave trading cart charging the enemy for glory !

Just started play aoe for the first time your videos have really helped me deal with my friends who have played this for many years thanks.

Seeing the enemy uphill "They can't do that! Shoot them.....or something"

U are the best! Spirit of the law, whem i see a fun video in youtube about aoe 2, its ever of ur channel. Thanx! From Bogotá, Colombia. New sub =D

I love your videos Spirit - I'm especially impressed by the high production values. I'm about to start trying to make KZfaq videos myself (not AoE). Do you mind if I ask what software you use to put them together?

The 33%/50% - rule of thumb is a nice one to work with, thanks.

Spirit of the law : is it worth it to fight uphill? Obiwan Kenobi : [visible confusion]

Perfect analysis. It goes without saying that sametimes bad fights resourse-wise can be game winning fights. For example, killing an army of mayan plumed archers with normal archers is good even if its not cost effective, because enemy can only rebuild the army from castle, while you can spam 20 archery ranges.

that Hi-res Intro!

1- With Cavalry Archers, I NEVER fight downhill. I just march them up the hill to fight. 2- If you are attacking elite skirmishers with crossbows from downhill. Start getting omega 3 supplements.

Hey Spirit of the Law, instead of numerically solving it, you could just have solved your equation with setting A_f = A_0 - B_0. You can solve this and get: A_0 / B_0 = 1/2 * (1 + \beta / \alpha) With \beta / \alpha = 1,25/0,75 you get A_0 / B_0 = 4/3 , which is exactly what you got with your grafic calculator, as you showed in the video. The advantage of solving it analytically is, that then you have a more general result, which can be used for more cases. I hope you will see this. But nevertheless, this was a great video! Keep up the good work! (I am sorry for my dilettante english; it's not my first language)

I hope you won't run out of ideas cause I love your videos

In general, with A on the hill and B below A, then if each A unit is z times stronger in attacking B than B is attacking A, then B will need to have (5z+3)/6 times as much units as A to get equal losses. Also, in order to win the fight, B will need to have at least [(1+5z/3)/(1+3/(5z))]^0.5 times as much units as A. This means if one's troops are two thirds (66.6666%) stronger, then engaging uphill is safe if one has the same amount of troops. If one has 50% more troops, it is not safe to fight uphill if the opponent's troops are 35% stronger.

Can't wait for your Cuman and Tatar civ overview

6:55 - I'm still feeling dread about fighting up-hill... And then you show a flurry of Chu Ko Nu bolts flying FROM both sides! 'Twill be nightmares for tonight I guess.

Fuck. I discovered Lanchester's square law by myself a few years ago and thought I had discovered something unique. I had no idea somebody else had come up with it first. The reason you square the army size is that you need to account for both hitpoints and attack. When you increase the army size by 30%, you increase both your attack by 30%, and the number the enemy needs to kill by 30%. That's why bringing your attack back up to even while the enemy does 25% more damage than you is balanced. You have 30% bigger army and they do 25% more damage.

*trebuchet flings a rock at a castle* Spirit of the Law: *squints* yeah I can calculate that

My president says as long as I have a bigger button than my opponent, then I'm good. XD Thanks for the video!!!

I didn't even know hills give an advantage! But it sure does explain alot. Amazing how old this game is and it had hidden mechanics like this all along. No wonder my dad always won. Bastard probably knew all of them.

Ooh, percentage based math. Something about it always appeals to me. I like the way SotL explains this too.

"Assuming you fell asleep at the first graph" hahaha you got me man, i m parallel with math. I don t even play AoE but i find myself watching matches like my father watches football lol. Ps. I m surprised I m this early to one of your videos.

Great content! I had that 50% in mind as a guess when the video started, turned out my intuition was right!

Quick bit of maths shows that for given attack advantages α,β and k as your weaker-army scale factor, you get a pretty simple answer: 2k-1 = α/β Plug this in for α=1.25, β=0.75 and you get the k=1⅓ answer which SotL found, but this works for **any** different strength army. For example, what's the value of a civ not having their last blacksmith upgrade? Well, the Halberdier has 6+4 standard attack and 3 melee armor, so if we compare that to 6+3 we have α/β = 7/6, and so k=13/12 or about 8.3%. So that means that the final attack upgrade for infantry makes your halbs 8.3% more efficient in a head-to-head encounter, at large numbers.

Okay, so is anyone else curious how armor/attack bonuses etc. play into hill bonuses? Are they added before or after the hill bonus?

Lovin' the content but when are we getting the top 5 petard civs vid?

You made me smile at the 3m 30s mark as a math geek myself. Stuff like the M.I.T. blackjack counting strategy I used math and hundreds of millions of simulations myself to literally find perfect blackjack (or more perfect) plays and counting strategies. I found a better basic one to +1 deuces through nines with faces and tens at negative one and aces at negative four and ran math and calculated about a 10% better rate of return playing this over the basic one they had. I don't really play much blackjack at all because I hate the game after running variance sets and then playing some. I have better ROI's where I use math in my profession of gambling that blackjack is a waste of time. I know I used a much different subject, but I do love math and computer programming.

The math, you are so good at it

Never thought i would need square root again after school but here we are

Congrats on bring back the old intro.

Thumbs up for the intro.

Great video keep up the good work!

@spiritofthelaw it seems like you did not include the "bug" where units that die before their projectile hits their target do full damage and remove the +/-25% damage modifier completely. I have no clue how it might be added mathematically, but i am certain it influenced that original 50% you were talking about.

IT'S OVER SPIRIT OF THE LAW, I HAVE THE HIGH GROUND

Classic SOTL video! I love these types!

So in order to beat Obi-Wan comfortably, we need another Anakin.

Finally, full intro, Definitive Edition.

0:40 that female villager can see the code!!!

I love doing it with the scout vs scout at the beginning of the game agahah

Spirit of the Law discovers the Unreasonable Effectiveness of Mathematics in Age of Empires

That math he does in videos is truly stunning 😅😅😅😂😂

Hi SOTL, Viper mentionned a while ago he was tempted to comment your vids, and said recently that after contacting you, it wasn't going to be a thing, and that you had pretty good reasons for it. I was wondering what those were? Keep up the great work mate

Great video!