Is it ever worth it to fight uphill? 

"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."

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

*Insert Obi-Wan Kenobi meme here*

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?

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

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.

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

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

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

The beautiful old intro is a gift from the gods.

The intro is back again! :D

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

Anakin tried and it didn't end well.

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...

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

The intro music

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

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

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

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

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!!

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

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

Great video keep up the good work!

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

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

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

Great video!

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

Classic SOTL video! I love these types!

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

God I love that intro so much. Never change

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

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

Great video!!!!!

Nice video!!

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

OMG THE INTRO!!

3:21 PERFECTLY BALANCED AS ALL THING SHOULD BE

Please do a video covering Turkish spearmen or other "unusable" unit lines like Persian swordsman. Let's we if they have situationally any use

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

wonderfull video, congrats

Can't wait for your Cuman and Tatar civ overview

amazing video

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

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

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.

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

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

"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!!!

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

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 .

It's over Anakin !

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%.

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

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

The math, you are so good at it

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.

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.

I would love to see some tests with melee units. It gets messy. One army starts with the high ground but the armies start to mix units with both sides having the high ground over enemy units. I imagine the army that starts with the high ground has an advantage that holds throughout the battle and especially in the first few seconds of the engagement but not as large as ranged units who always maintain the high ground.

Dude, nice!

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

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.

The best intro

that Hi-res Intro!

You're welcome, thanks for making the video. :-)

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

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

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

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

2:37 so if I choose to not square root at the end, I can greatly increase my army size!

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

Thumbs up for the intro.

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

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

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.

Oh my god the intro!!

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

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.

According "The Art of War - Land Battle" , it's a good strategy when using the infantry.

You had the perfect chance to use the meme

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.

I'd love a vid about monks in general, how does faith work? Is healing worth it? And stuff. They've always been a mystery to me

Extrañaba esa intro

it would be awesome to test the cost of rushing up the hill to take a fight with someone holding it. Coz that's what people usually do to clear hills, and I feel like its more cost efficient than, for example, the fight at 6:00 (a few exceptions of course like high dps units). But obviously it will depend on how much damage you take before you get on even ground, ballistics/thumbring being a huge factor.

Have you ever done a top 5 navy civilizations video? I am asking because I want to know how to get a good navy to help my friend in his Huskarl rush

@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.

Finally, full intro, Definitive Edition.

Anyone know if the size of the hill matters at all? Do you only need to be 1 'elevation point' above the enemy for the buff to occur or is there some sort of sliding scale? Bigger hills == Bigger bonus? There was some cliffside bonus too... does that stack with this also?

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.

Math: It's a beautiful thing.

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

Could you do a follow up to this on structures on hills and how it works? I have never understood if my castles benefit from up hill advantage and if they do how it works with things like a siege ram that has the frontal line of damage. Also how does it work when my castle is built on one tile of hill but every other tile is not on a hill?

I smell the Tatars overview coming next. :))