The Mathematics of Banana Farms (Bloons TD 6)

[@alexonscience]

TLDR: I made a small mistake in my calculation for chi, the best farm is actually Monkey Bank with greater production, my video is not comprehensive but more a demo of applying math to everyday situations, and Valuable Bananas is still good as a crosspath. I have a creator support code now! alexos Correction: Looks like I messed up the formula for chi, the income generated by a bank in delta rounds is delta * g, not just g, so the banana compound efficiency should be chi = (1 + delta * epsilon)^(1/delta) - 1. This should not affect the Monkey Bank Theorem (though makes the unequal case harder to prove) or the general shape of the graphs, but they are flatter than what I showed in the video. I updated the spreadsheet, so Monkey Bank is back up there in terms of banana compound efficiency, Monkey Bank with greater production is actually the highest at 7.41%. Thanks @r-prime for pointing this out. So it's ironic that I used the Python script to prove my formula was correct, when my Python script had the exact same bug. Which is a valuable lesson: When you write tests or design experiments to verify a conclusion you already made, you are often biased to make the exact same mistakes in the test causing it to reinforce your incorrect beliefs. Also, there are so many other things you could explore. I only looked at Medium difficulty with no Monkey Knowledge, special abilities, special effects, or synergies. For example, ignoring "discreteness efficiency" is a big assumption: Even though Valuable Bananas has terrible banana efficiency, is it better to get it when you can afford it instead of saving up until you can afford a second banana farm? It's important to keep in mind that simplified mathematical models are often useful to consider but easily break down in real cases. This is the same reason economic theories often fail to model the real world accurately at all, too many simplifying assumptions. I encourage anyone interested to try relaxing various assumptions to see if the optimal farm changes. The point of this video isn't to provide a definite answer, but to show the approaches you can use to solve similar problems in your everyday lives. (And to clarify Valuable Bananas is still good as a crosspath, just very inefficient as 0-2-0.) Google sheet with all banana farm data: docs.google.com/spreadsheets/d/1Agt4FBDX6IYRpjaGFumrn9OL9ttx8UxPpw2lgE98re0/edit?usp=sharing

[@lol-yz8fh]

make a vid calculating the percentage of shots Quincy misses based on different types of Bloons

[@niffthesniff8258]

Please make more videos going into more specific and niche cases, I would really enjoy that personally

[@lool8421]

it's also interesting how the complexity of farming is high enough that common sense is sometimes actually a better use than calculus to find the most optimal solution because obviously you get different stuff with different tower setups and maps, as well as tower restrictions and difficulty changes

[@feliciadiegottin6759]

Do you have discord I think I could show you a way that would have saved you a lot of work with the spreadsheet

[@youvegoattobekittenme6908]

Yeah, I think with pruning it's much easier and more accurate to compute the fastest way to reach a specific money threshold purely through farm income starting with a single farm and 0 money.

[@normalhumanperson8858]

still died on round 63

[@_chirp_6108]

use glaive lord or spirit of the forest :)

[@tayty1238]

Skill issue 😂

[@ThiagoGlady]

​@@_chirp_6108 Yeah but not everbody plays on begginer maps sorry

[@schmeedorcocloa1986]

@@ThiagoGladyboth of those towers remain good in expert maps (especially with the recent glaive lord changes) and excel at the intended purpose of demolishing r63

[@ThiagoGlady]

@@schmeedorcocloa1986 That's true, but in Experts map you don't have cash to spare like that so you can buy a Spirit of the forest on round 63 without demolishing your defenses for later rounds

[@kulrigalestout]

"Unpaid monkey interns" should be implemented in game, even if it's just a visual of a monkey scrambling around with cups of coffee

[@ponocni1]

Thats starter dart and after you make xx4 village, another dart every round.

[@a.person1805]

new Monkey Wall Street pet cosmetic please I need it

[@purplejerry1]

🅱en

[@Steampunk_Wizard]

I always thought it was on brand to name my Dart Monkey "Unpaid Intern", because of the free MK.

[@mhmedgamer1]

لايك رقم الف

[@dominorborg605]

So this is what Benjamin is doing on that computer for 100+ rounds to make the banks better

[@alexonscience]

I love this lol

[@jamoin8188]

Every one is talking about the Gender pay gap, but finally someone is addressing the Banana Payback

[@EeveeRealSenpai]

​@@jamoin8188 Civilized monkeys discussing banana inflation and prices. 🧐

[@hanasong8296]

The irony of this is that Benjamin should be sold around round 80 because his siphon ability downgrades the MOABs and in late game the cash he generates does not makeup for the loss his attack does. In high level play, when the map allows for it, Benjamin should be placed behind an obstacle to obstruct his attack from hitting.

[@ryanhodin5014]

​@@hanasong8296In lategame these days Ben makes shitloads of money (at max level he makes $5k per round, plus skimming $2 per bloon spawned - compare to $5400 per round for the 025 Monkey Wall Street, which Ben will absolutely outclass on something like Round 95). Plus, thanks to freeplay scaling, Ben's Trojan very rapidly stops costing much money. Ceramics and below are profitable to trojan on Round 81 if you get level 16, and MOABs and DDTs both become profitable to Trojan on 120 (and are costing you increasingly less as you approach that). By 120 even trojaning a BFB only costs you about $10. So, really unless you're using a very stally strategy, you probably still make money keeping Ben wide open lategame. Putting him where he can't see the track is strictly better, of course, but even on Round 96 (the most profitable round in the game) Ben would have to cost you over half the money from pops to be worth losing the end-of-round income plus skimming.

[@lolypopmaster11]

23:01 this is where this video starts for chimps players

[@alexonscience]

Very true LOL

[@beanie1694]

Don’t forget benjamin

[@garfled23]

Ty

[@nated4237]

My dumbass clicked the time tag ☠️☠️☠️☠️☠️

[@killmario2092]

Really helpful

[@somewhatcoolguy2196]

This is the perfect kind of content on youtube. No overly-saturated thumbnails, No big names or money, no annoying sound effects constantly. Just simple content like this is so refreshing.

[@alexonscience]

That's very kind, thank you!

[@Vlek]

@@alexonscience ohyeah the thumbnail just made me go now THIS is what they shouldve shown at school for advanced math to get students to know why they need it do homework for no reason? hell no understand vast advanced math for bloons game? y e s

[@alexonscience]

@@Vlek Absolutely! At school they often just show a definition on page one and you're like, but why tho? If you motivate the concepts first with concrete examples it's a lot easier to understand

[@Vlek]

@@alexonscience coming in with a funny and actual true reply too? damn your channel is gonna be big if youre this good at what you do

[@TheFreakyFish251]

Nice pfp, memento mori

[@idklol309]

You know its good when its a 23 minute video on how to make the most digital money from tiny little silly banana farms to pop silly different color bloons and blimps for the monkeys.

[@alexonscience]

I don't know what you're talking about, this is a very serious video 👀

[@peestrem31]

Monkey business is no laughing matter.

[@alexonscience]

@@peestrem31 probably should've included monkey business as a joke somewhere, missed opportunity. Also very surprised there is not an upgrade called that

[@peestrem31]

@@alexonscience yes you should have

[@villainous69]

It's not silly. It's not JUST a banana farm......

[@bantix9902]

The field of BFT (Banana Farm Theory) is just emerging, you laid down some foundational theorems which mathematicians can build upon. Great Video!

[@alexonscience]

This is my favorite comment so far LOL

[@vinesthemonkey]

it's called accounting

[@imafirenmehlazer1]

Can you imagine if the devs actually start trolling 😂 with players changing the balance of farms when ppl get close to figuring it out LOL

[@ShutUpKetchup]

​@vinesthemonkey just need to calculate ROI

[@christiangray7826]

I love how you talked math for 21 minutes and summarized it to “valuable bananas is trash”

[@alexonscience]

The most important takeaway

[@vlad-darius-2008]

@@alexonscience that's interesting, because in btd5/battles1 valuable bananas is actually really good

[@shivinpokemon]

Never buying valuable bananas again

[@marxunemiku]

i assume he's talking about 020 farms, which nobody buys any way

[@OctopusGamingRoblox]

@@marxunemikuunless getting a deposit bank

[@parallaxninja456]

same haha

[@alexonscience]

@@marxunemiku Yeah I guess it was a bit misleading, as a crosspath it's not bad but 0-2-0 is the least efficient farm

[@ponocni1]

Its worst starter, but best crosspath in long run. Not worth touching until T3 maybe even T4. MK makes it even more sexy.

[@lool8421]

there's a much more difficult problem: you have 1-0-4 and 1-0-5 village, as well as boat farming synergizing with xx4 farms which makes the efficiency vary, then ability farming is just a different story because it heavily depends on the rest of your setup, druid farming also relies on the amount of available space then there's also the fact that if you can't afford farm A, you can use leftover cash to buy farm B instead which would be more efficient than waiting 1 more round additonally banana central buffs other 4xx farms and farm selling is a thing basically as much as math points in the right direction, it's nearly impossible to say what's the objectively best farming method

[@SederixHD]

Yea and overclocking farms is a big deal

[@lordscorcher]

this video is a start of banana farm rabbit (or monkey) hole

[@alexonscience]

Totally agree, I made a lot of assumptions in my video. Certainly wasn't aiming to provide a definite answer in this video. I added a pinned comment clarifying.

[@GeshemJR]

it is possible to say what's objectively the best farming method, just look at how optimized farming is in top level play, where they can squeeze out every last drop of possible income for every situation

[@dustinbragg1921]

​@@alexonscience I can't help but wonder just what kind of horrific rabbit hole we would send an economist or financial expert down if we put this theoretical situation with all these options in front of them and asked them to solve for the most logical conclusion for say making it to Round 200 with as much money as possible or as much cash generation as possible. Diabolical.

[@crackhouse_slag]

im convinced he just wanted to flex on us at 3:26

[@TheMonkeydela]

Clicked for the Bananas Stayed for this man trying his hardest to convey BTD6 logic to a wider audience

[@ozthebeeman]

even tho this video is great from a math stand point, in actual btd6 banna farms are pretty much solved. ofc there are some tricks to earn more money with certain tricks, but that doesnt really matter here. ever since boss bloons were added, people really got good at getting max value from farms, and the most popular strat used is: start with 4, 2-0-0 farms, turn them into 2-0-3 farms, sell all of them to get monkeyopololis with 1 0-0-0 farm (or2 2-0-0, but it wont add much) and then just go 4-2-0. you want as many 4-2-0 as you can fit inside the range of monkeyopolis, this changes based on the map and hero you have. go for 5-2-0 and an overclock, and overclock your monkeyopolis and 5-2-0 while spamming as many 4-2-0 as you can fit on screen. ofc there are other strategies, some include boats, but this start is easily the best and works on all maps (for boss events some maps have no space for farms, but they arnt used for boss events for this exact reason).

[@alexonscience]

Totally agree, I made a lot of assumptions in my video. Certainly wasn't aiming to provide a definite answer in this video. I added a pinned comment clarifying.

[@vinesthemonkey]

for t1 boss usually you need a whole lump sum of money. the way I did it last week is 202 and 220 farm, sell 202 for bank, get another farm up while bank is accumulating, cash out bank to get imf, then collect / deposit around $8700 to maximize interest. and take out the imf loan for t1

[@ozthebeeman]

@@vinesthemonkey I hope I can explain this well, but basically the idea is that 2-0-3 sells for an increased amount plus the increase in money per round, means you end up making your money back way faster than any other farm method in the early game. I don't remember the exact amount, but I believe it's around 3 or 4 rounds. To my knowledge this is easily the fastest way to reach monkeyopolis which trumps every other farm option. There is a reason they call it meta....

[@lesbianmortis1261]

@@ozthebeeman I dont get something, you upgrade the monkeyopolis for it to absorb just one farm with no upgrades? why do this? the income of just one unupgraded farm just nothing? is there any secret buff from the monkeyopolis? im very confused

[@ozthebeeman]

@@lesbianmortis1261 to my knowledge this is the best cost effective early farm. And waiting for 4-2-0 means you are making less money. Going early monkeyopolis means you get more early cash to invest in more famrs,then you can sell the monkeyopolis for a maxed out monkeyopolis.

[@jstrategy1744]

Finally, some math that I will actually use in everyday life.

[@normalhumanperson8858]

the cuts are too slow for the attention span of an average youtube viewer in 2024, but the quality is outstanding very hbomberguy-esque

[@alexonscience]

That's what the playback speed is for LOL

[@beaverbuoy3011]

@@alexonscience true!

[@kiprasking8581]

@@alexonscience Big brain

[@thatdude9091]

@@alexonscience2x play back speed = half the retention. Even if you watch it on 2x speed, the algorithm sees it as you’ve watched the video for half the time, making it want to promote the video less

[@boi2192]

Dont suggest downgrades i beg u

[@therealorangemangoes]

bro has the kurzgesagt book in the background 😭

[@alexonscience]

I still haven't started it lol

[@C_ook.ie_]

​@@alexonscience you should read it its really good and interesting and the pictures are pretty

[@yevhenlos6255]

@@alexonscience At least tell us you've gone through "What If".

[@soopwizsoopwiz343]

The Dune Ornithopter in the background shows you are not just an intellectual, but also a man of culture.

[@ahmeteminsenyurt131]

nuh uh im just going to spam marketplaces and you cant stop me

[@alexonscience]

Personally use them the most, they have a pretty high banana efficiency, but mainly I just don't wanna manually collect the bananas

[@joshholmes322]

We need a laziness theorem

[@pstm53]

@@alexonsciencemarketplaces also buff the bottom path buccaneers, not sure if it’s worth it but I use them a lot

[@wilale]

Get middle cross path, monkey knowledge buffs it more than top

[@Catman_CM]

As far as the kind of farms you can expect to get up with enough time to make a profit in a map like Impoppable, 2-0-3s and 0-2-3s are only outperformed by 4-2-0s, and X-X-3s are more spammable before you reach the map's space limit. I suppose selling a Marketplace to replace it with a 4-2-0 later would be a good idea, but keep your Rho value in mind before doing so!

[@quinneady4524]

Just looked at the spreadsheet and there was almost 15 people active on it, proving this is exactly what we need. We don't need tutorials and best strategies, we need maths!

[@wtfox8206]

This is simultaneously the best economics lesson, math lesson, and bloons td 6 lesson I've ever seen

[@createeel99]

placing banana farms to that degree only scratches the surface of carpal tunnel. As an avid geometry dash player, carpal tunnel is like food air and water.

[@alexonscience]

I laughed out loud at this LOL

[@AverageBloonarius]

whadda bout a pikmin player

[@Octagongd]

I have over 1000 hours in that game and I quit the game because of the insane hand pain I was experiencing.

[@mr.chicken_gamer]

Find which is better grinding for monkey money or working a minimum wage job to then buy the monkey money

[@blockybeanz9924]

Should do that irl…

[@tinyturtle1898]

It actually better to take out a massive loan and avoid paying it back

[@mr.chicken_gamer]

@@tinyturtle1898 ah yes how could I forget

[@erierierierierie]

i left a reply but i realized i was just insanely wrong with the math. working a job will give you more monkey money lmao

[@mr.chicken_gamer]

@@erierierierierie Would it be faster though?

[@GameJam230]

An important thing that I think a lot of people disregard in calculations of the best banana farm is this; you can reinvest more money than the banana farms themselves produce, because you MAKE money from regular gameplay too. This means the true best banana farming strategy is not just determined by the money it will generate in return for investment, but ALSO the money you generate through natural gameplay (minus the minimum amount required to survive the rounds). Since the money earned per round is wholly dependent on the round number, and not affected by the number of farms you have, it would likely be better to treat the extra cash earned per round as a OFFSET for the price of a banana farm purchased NEXT round. So, if an upgrade would cost you $700, and this round gives you $500 in extra spending money, you could look at that as the upgrade only costing you $200 NEXT round. Now you'd need to determine if the money saved by waiting a round is greater than the amount it would have produced if you upgraded THIS round. If the upgrade would cause the banana farm to produce $550, then it is more worthwhile to buy NOW, as you would be $50 ahead of if you waited.

[@ThiagoGlady]

Yeah but how would this fact change about the farms efficiencies? It really shouldn't matter if you earn money per round or not, a good farm would still be a good farm and a bad farm wouldn't be better because of it

[@GameJam230]

@@ThiagoGlady Because we're talking about MAXIMUM efficiency of money earned here, not just about which farms are good. That means I don't just care about which farm crosspath earns the most money per round, I care about the exact moment I choose to buy it too. This is a highly relevant subject for bosses which often have elaborate farming strategies like buying multiple farms before upgrading any of them, making Monkeyopolis villages with some amounts of farms in it, and even selling the farms at certain points. Being able to factor in the amount of money you make in regular gameplay is CRUCIAL to getting a perfect optimization, since as I said, waiting a round could be the difference between just $50, but what about making entirely different decisions? What about instead of upgrading, you purchase a new farm? What if instead of buying a farm, you buy a Moneyopolis? What if you additionally factor in selling the farm at a certain point? None of these things are accounted for by just looking at which farm upgrades get the best return, which is why the meta boss strategies don't make sense if you only calculate their value in terms of money per round minus investment cost.

[@arcturuslight_]

​@ThiagoGlady it doesn't have an effect in the type of calculation presented in the video, you are correct. I think that commenter is talking about more detailed real strategy calculation, where granularity is taken into account, we can't take limits, and we have to take into account the income produced by low tier farms on their way to upgrade to higher tiers. There, the upgrade paths and whether you do any selling depends on the rate of non-farm income.

[@GameJam230]

@@arcturuslight_ Yep, pretty much. If all we cared about was the cash per round, a simple Google search will do well enough to find out the highest cash per round and you can derive repayment time from that, but an actual analysis of farming strategies requires far more than only caring about what they produce.

[@alexonscience]

Yup there are so many other things you could explore, I only looked at Medium difficulty with no MK and no special abilities or special effects. But ignoring "discreteness efficiency" is a big assumption, e.g. even though Valuable Bananas has terrible banana efficiency, is it better to get it when you can afford it instead of saving up until you can afford a second banana farm? This is the same reason stuff like mathematical economic models often fail to model the real world accurately at all, too many simplifying assumptions. I encourage anyone interested to try relaxing various assumptions to see if the optimal farm changes. The point of this video isn't to provide a definite answer but to show the approaches you can use to solve similar problems in your everyday lives. But imo, still banana efficiency and banana compound efficiency are useful for consideration even if they aren't everything.

[@JackSmith-oq4mf]

0:27 ninja kiwi does say lore about game whats end of track r monkey towns bloons try invade

[@dustlicker4]

always the ddt and bads eating everything up

[@Gamesux421]

One of the most important bits about the farms is selling. It will have payed for itself very quickly if you keep selling in mind, and a bunch of 420s and 1 520 is eqsily enough to buy any paragon if you sell some farms after theyve been making for a little bit

[@crackersphdinwumbology2831]

3:15 1000 hr pro btd6 player here. There's a lot of context I'm not sure you could weave into the math. Banana Central improves the income of all Banana Research Facilities. It is also affected by Overclock, Monkey City, Geraldo's fertilizer, and Valuable Bananas. The "best Banana farm" is also impacted by your map type and what round you plan on selling your farms. I.E, is a higher ROI better when your farm isn't going to be sold for 70 rounds? As for banks, it might also be important to include Benjamin buffs to your bank, and you can make deposits to tier 4 banks. This is a fantastic video. You should do some boss challenges and test these theorems.

[@Fritz-co4pb]

Got any advice for a noob? Especially on getting monkey knowledge.

[@alexonscience]

Totally agree, I made a lot of assumptions in my video. Certainly wasn't aiming to provide a definite answer in this video. I added a pinned comment clarifying.

[@crackersphdinwumbology2831]

@Fritz-co4pb For obtaining monkey knowledge, your best and most important move is to Black Border all of the Beginner maps. This means completing them once on each game mode of each difficulty. This will also give you a healthy income of Monkey Money to spend on heroes which will be important for BBs on Intermediate maps. Geraldo, Psi, Striker Jones and Etienne are the best options for many maps. CHIMPS and Impobbable give the most EXP towards your levels as they last 100 rounds, they also provide the highest Monkey money. Experiment with a lot of different towers. Don't shy from abilities, especially low cooldown abilities. Happy hunting!

[@crackersphdinwumbology2831]

@@alexonscience Of course! I really enjoyed watching you, and you're clearly a skilled mathematician. It's almost crazy how complicated economy calcs can be during boss challenges. I enjoyed watching quite a bit!

[@imadudewithissues]

​@@Fritz-co4pbfree dart monkey will carry you every game mode that's not chimps

[@JoaoPauloStawiarski]

when i studied calculus for the first time, my fun was solving problems that i create involving games or others bizarre things. watching this video makes me realize that this is the fun of maths, not just solvin the fucking Thomas's or Demidivich's problems, the college forces you trying to be like those guys and just destroy the real beutty of learning. congrats, you just earn a subscriber.

[@staotheduck4853]

I was looking for this a week ago but couldnt find it anywhere and didn't feel like figuring it out. Thanks for giving us practical uses for math, it makes learning so much easier!

[@alexonscience]

Glad to have helped! I very much prefer introducing concepts when the need for them arises naturally, instead of definitions being thrown at you out of nowhere

[@the1barbarian781]

Worth noting that using Ben and having bank monkey knowledge helps banks too. Great video Edit: Also some farms have better sell back which helps them. So many tiny details in something that would be seemingly simple

[@justanotheraccounthere2014]

Exactly, the best banana farm depends on your case. In rank bossing you will buy 200->202 for the extra sell back value to rush 230. This is how a game on monkey popping bloons is the paragon of tower defense games.

[@firytwig]

It also doesn’t take into account village buffs, monkeyopolis’s, the banana central’s global buff to all BRFs, and whatever imf farming is

[@alexonscience]

Yup there are so many other things you could explore, I only looked at Medium difficulty with no MK and no special abilities or special effects. But even then the "discreteness efficiency" is a big one, like even though Valuable Bananas has terrible banana efficiency, is it better to get it when you can afford it instead of saving up until you can afford a second banana farm? This is the same reason stuff like mathematical economic models often fail to model the real world accurately at all, too many simplifying assumptions. But still banana efficiency and banana compound efficiency are useful for consideration even if they aren't everything.

[@firytwig]

@@alexonscience from testing, valuable bananas specifically is only worth getting if it’s a crosspath on a t3+ tower, or when saving for a bank you might as well have the bonus income. It is not worth buying alone instead of saving for another farm because good lord it is ass. Getting valuable bananas is barely cheaper than another full farm.

[@tinyturtle1898]

@@alexonscience 0-2-0 Valuable bananas is obv not worth it, but 2-2-0 is sometimes worth, if you dont want to save up for a second farm

[@HotAnimeWaifu]

haven't watched it yet, but i have a strong feeling this will reignite my love for math like no other video has

[@chumbucketpatty]

I love math, I love BTD6, and now I love this evening! thank you so much. Keep making videos, please.

[@trevas1047]

I'm surprised that there were that many limits. I guess this is only the surface of banana farms and the rabbit hole goes much deeper, especially considering boss events. Great video!

[@Crockbucks]

this video is really fun to watch thank you pls keep doing stuff like this its really enjoyable and shows that there is still hope for quality content in yt

[@tannerwolfe3349]

This is the most genius, educational video, in the history of Earth.

[@alexonscience]

As a completely unbiased party, I agree

[@Rabid-Bunny]

00:18 you’ve convinced me

[@dylanirt3905]

I can't believe how enthralled I was by this.

[@ATOM-vv3xu]

050 is very strong since you can stall the rounds to get 2x10k per round from the ability, banana Central buffs other banana farms and central markets buff ships, you disregarded monkey knowledge and banana farms are not alone, there are also villages, snipers, druids, boats, Helis, Alchemists, abilities, engis, heros and probably more that I forgot

[@swawerq2564]

Yes the premise of the video seems to solely focus on monkey farms and nothing else. If we were to include other forms of cash generation and discounts then here's a list to start off with. Towers besides farms that can increase cash generation - Sniper middle path - Buccaneer bottom path - Heli middle path - Submarine top path reduces all cooldowns, boosting efficiency of cooldown based cash generation - Super monkey top path - Alchemist bottom and possibly top path - Druid middle path - Village bottom path - Engineer middle path can overclock villages and farms, bottom path Heroes that can increase cash generation - Obyn with his wall of trees - Benjamin - Geraldo with his action figure and fertilizer - Corvus at max level with his nourishment spell Monkey knowledge - Military tree has trade agreements, military conscription, charged chinooks, advanced logistics. - Magic tree has speedy brewing possibly giving faster lead and rubber to gold. - Support tree has tons of farm and village knowledges. - Especially the discounts, more valuable bananas, bigger banks, backroom deals, inland revenue streams, bank deposits, global ability cooldowns. - Hero tree helps your hero level faster with some upgrades. Other ways to increase cash generation - Stalling for ability based cash gen - Regrow farming (Smart Persan made a video recently on the topic) This list may not include every method possible but these are the general things I would consider first.

[@LetsPlayCrazy]

21:08 "disregarding abilities"

[@wrc2933]

05-2 let’s u resell it better too

[@alexonscience]

Totally agree, I made a lot of assumptions in my video. Certainly wasn't aiming to provide a definite answer in this video. I added a pinned comment clarifying.

[@tinyturtle1898]

I used to build a bunch of IMF Loans to go deep into debt to afford the Tier 5 Monkeynomics

[@heheheiamasupahstarslam5397]

ultimately the sheer inefficiency of valuable bananas is worth it as banks are very effecient

[@heheheiamasupahstarslam5397]

or if going top path, the 30% more money from the crosspath pays for itself after at most 11 rounds with a 3-2-0. 6 extra per banana, 16 bananas a round = an extra 96 cash so make like over 1000 and its paid off and every round after that profits while other crosspaths offer no profit

[@Pyro_Manac]

Interesting idea: you can use linear programming to try and solve this problem. I'm so curious how it would work i'm tempted to try it myself. You basically set the objective function as what you want to maximise. It consists of a variable to keep track of each type of banana farm and how much money they make during a round. Multiplying these together and adding them up gets you your objective function. The constraints for the model would consist of what the maximum amount of banana farms you can place are, the effects that certain banana farms have on other banana farm's income (eg. The banana central giving more income to all other BRFs) etc. I don't really play bloons anymore, but this interests me enough to want to get back into it. I guess engineering is a useful skill after all.

[@Pyro_Manac]

If I ever get around to it, i'll send the link to the maximisation problem and see if a person can solve for the most effective income for each price bracket.😊

[@jacobstrength3945]

You need to make a community of some kind to do more research on this problem. I absolutely love optimization like this and am willing to dedicate my brainpower to the field!

[@mangcrow5738]

literally the first time I fully understood the point of limits

[@billystorm5347]

I truly love how the most gamers use math for is random optimizations in oddly specific games that rarely ever have anyone apply the calculations. Also great video I always loved following people doing complex math.

[@LnlyCloud]

"I feel like every sentence in this video is a sentence that no one has ever spoken before" SO TRUE lmao this was a wild ride but a great one. Love the video.

[@alexonscience]

That's very kind!

[@braydensry3735]

I love the simplicity and special stuff you did in the caption, those little messages are always funny. Great video!! (I needed a video to convince my friends why monkey bank is the best)

[@Krytz3n]

Ngl this along with pizza size are my fav ways to use maths

[@lassestewart578]

Pretty amazing mathematical approach!! We could also turn it into an optimization problem by focusing on generating the highest amount of money by the final round, considering the starting cash from each difficulty and the bonus cash per round (round R's RBE plus the extra 100 + R at the end), and considering the game mode's constraints (MK, difficulty settings, etc). We could split this new problem into two by assuming all bloons are popped regardless of our defences vs taking the current record for most optimized run in that given setting (with the fewest heroes and upgrades).

[@davidd.w.8681]

2:16 Like any serious math video, time to watch it at 2x speed for the complete experience.

[@justanotheraccounthere2014]

Very nice video. We can take some educated guess into the realm of late game free play/boss event (space limited with many rounds to go) farming strategy and early game (discreteness) farming. Space limited: We are not buying any new farms, so efficiency does not matter. We do not need to worry about paying back the cost of farm since there are many rounds to go (e.g. 80-120 in boss events) and a faster linear increase will always take over eventually regardless of the initial difference in leftover cash. Therefore, to maximize income long term (e.g. at the end of round 119), we want to pick the highest income (g). Sorting by that column, we can notice that all T5 are good options and the best non-T5 is 420. Well, that is exactly why during boss events on land maps, you spam 420 and almost always all T5 farms. Early game farming: It is a balance between cost of the farm itself (afford it as soon as possible) and its efficiency (maximize investment return). Therefore, those with low cost but high Banana Efficiency stands out. Notably, 230 has the highest Banana Efficiency but relatively cheap price, so this is a good target to aim for early game. In fact, 2x200->202+220->sell 202 and upgrade to 230 is developed because of the attractiveness of 230 in early game. BTD6 farming is a discrete optimization problem with many choices between rounds and finding the most optimal solution is computationally very hard. Genetic algorithm might be very useful here, if we consider the "population" as different strategies, "genes" as actions taken between rounds, and "fitness" as the leftover cash plus sell back value of all farms.

[@moose200]

this video is peak, but whats even peaker is the little plushie moose on the books i love him i love him i love him anyways im suing you for using my likeness

[@alexonscience]

lmao It's the unofficial mascot for an org in my company, I love him as well

[@AverageBloonarius]

I WANT MOOSE

[@jumpingaxe3373]

I love how all of this will likely change entirely in just a couple weeks of balance updates.

[@timnijs1278]

I think the idea from the developers behind having Banana bank and IMF loan being worse is that it works passive. Thank you for the whole video, I'm certainly going to use the spreadsheet and play around with it a bit.

[@xspearozx6486]

Oustanding! I used to think I was bad at math and failed calculus, but this video was easy to follow and reinvigorated myself in realising I am actually good at math.

[@alexonscience]

That's great to hear! It is usually easier to understand concepts when the need for them arises naturally instead of them being thrown at you out of nowhere. Introducing limits because we don't want to deal with remainders grounds it much better than showing the definition on the first page imo

[@n_kas5812]

i feel like ignoring monkey knowledge is kinda silly cause people who care about this kinda stuff already play the game a lot and probably have the important monkey knowledge like more bank storage but this is a great and very high quality video, i like it a lot!

[@Jyroson]

Well the real end-game is chimps mode anyways right? Which has no MK. And if it works in Chimps, it's probably better in non-Chimps.

[@n_kas5812]

@@Jyroson um...

[@n_kas5812]

@@Jyroson the I in CHIMPS stand for (no) income

[@alexonscience]

True, but I think you could easily apply the updated numbers with MK for knowledge that just increases the stats. But taking into account synergies between towers is more complicated

[@sasho_b.]

2 things: opportunity cost and selling liquidity. And also Banana central income boost but ignore. Opportunity cost: any farm upgrade requires its predecessor to be bought. Any formula for it will have it as 0 when initial money approaches infinity, as we can just buy any farm we like. Selling liquidity: it is in some cases more optimal to buy a few cheap farms with good efficiency, and then sell them for an expensive farm with good efficiency, skipping over mid farms with mid efficiency. Specifically: greater production->bank, marketplace(xx3) into banana research facility, banana research facility->banana central. Ive not a clue as to if its relevant besides marketplace, if its separate from opportunity cost, or any of this. Honestly, someone should set up a graph search for what is the most optimal banana farm strategy, for any and all scenarios (i.e. money on a round may have factors like loss from defense or gain from selling). Sounds like a traveling salesman problem where the only true algorithm is to just check everything, or else to make a 90% accurate estimation.

[@forevermasteri4330]

Would selling liquidity apply to races too? -- To skip tower upgrade combinations that have a low DPS ceiling per dollar spent?

[@flex4711]

Using a 2-4-0 farm you can add your cash into the bank, that way you can get great interest every round, on top of the production. (I usually tap thrice to fill it to 10k$ as its limit is 12.5k, then it generates a lot of interest in addition to the production, without overflowing) In addition to that the loan ability lets you expand your banana empire really quickly. It would also be interesting to see how the other cash-generating monkeys like the sniper, druid, merchantman, etc. compare. Keep up the good work!

[@blacklight683]

"math is useless" -said no person with a brain ever how will i watch people analyze 50pages of code,13pages of research and 20pages of math to do 0.5more damage???? oh yea also computers and internet and building and all of that other minor stuff

[@rosesolomon7276]

I didn’t know I needed this video until it popped up on my recommended😂 I have a masters in math and love btd6. Glad I can skip doing my own math now cause I know I would have ended up doing pretty much this, though definitely not quite as thoroughly. Thank you for this!

[@droal.]

True BTD 6 gamers know, that this is just the first lecture out of the Bachelor of optimizing Bloons

[@overtimechaos]

Never had i needed a video to explain which is the most efficient banana farms

[@masterandexpert288]

THANK YOU for finally doing a deep dive on this. I've wanted to know about how to approach this generalized problem.

[@hahalolxyz0685]

This is the greatest video I didn’t know I needed to watch completely. 😭

[@alexonscience]

Glad to help!

[@riich.586]

Bro i just wanted a quick answer to the question, now im having WW2 flashbacks holy!! dude popped off. You could be making that up and I would still believe you tbh

[@banter2143]

15:20 "That makes sense!" Yeah man I'm gonna take your word for it 😵‍💫

[@ugthakthedestroyer4566]

i have terrible news, this video rings true _without taking into account the skill tree upgrades which can lessen tower cost and increase production and value_ . fantastic video. would love to see more btd6 math vids.

[@thomasandriessen1046]

If you're using compound efficiency as a metric then banana banks become the more liquid option. If for example you needed x amount of cash by round 40, then you can take out money that was compounding out of the bank right before the round without selling a farm. This can also be an argument for the bottom crosspath because they are able to sell for more of their value. It might be interesting to make a metric that tells you the factor of money you can have after a set amount of rounds. (allowing selling)

[@jaredking923]

Great video. I think there’s a ton of potential, content-wise, in deriving theorems from video games. Imagine doing this same analysis for Cookie Clicker with all the variables in that game.

[@ShrekOgrestone]

This video combined two of my favourite things; gaming and mathematics. Thank you for the awesome video kind sir, this gained you a subscriber.

[@TheVaporater]

Ill be honest...i can barely do multiplication very well, let alone understand ANYTHING in this video 😂 but it is absolutely fascinating to watch your brain work so magnificently! So even though i couldnt follow anything(i think monkey banks is the right answer? Lol)you said, i still watched until the very end just because its so impressive!!! I truly wish my brain worked even just a fraction like yours does!!!

[@Purin1023]

This video really shows just how insanely complex video games are. We basically need to ignore something like a dozen factors that are present in the game just to get these formulas to make sense. Great stuff!

[@Maker0824]

I’ve been wanting people to look into the random math of various games. Very glad I got one of my wishes fulfilled.

[@joegurt9053]

Wow, I got to re-experience losing my attention in math class and then tuning back in when the board is filled with variables I don't know.

[@NonJohns]

i was looking for a definitive answe but your approach was definitely nice as "math that i assumed was right and finally have confirmation for"

[@skyblockgrindertime3889]

bro managed to used binomial expansion

[@RonZhang]

What gets even more complicated is when you take into account of money generators and swapping normal defensive towers with Bloon Traps, Merchantmen, Rubber to Gold, and so on.

[@manveensingh3494]

The way you present yourself and your data has its own charm. Your hand moments, your way of speaking

[@swawerq2564]

This just goes to show how complicated btd6 can get! Great video!

[@NotJim64]

Instructions unclear used merchant ships

[@theinkingskeletonsmess]

This feels like a reality TV show for kids and they're acting like the kid is actually Part if the show

[@cyborgbob1017]

I think the TLDR of it all is that 230 farms best for efficiency(tho marketplaces are more affordable so sometimes people go for that first instead), while BRFs are best for when you already have a decent chunk of change and just wanna keep pushing your banana output to the moon without much regard at all to price efficiency. The actual boss meta reflects this too.

[@babayaga7842]

very very nice and refreshing take on a super already explored theme.... I was surprised when I saw your channel doesn't have thousands of subscribers!!! +1 subscriber, keep it up man!!!

[@Empanada__]

You have no idea fow how long I've wanted a video explaining the math behind banana farms😭😭😭 Thank you so so much🙏🙏🙏

[@mongooseV]

I love XKCD's what if?, it's such a great book and seeing it on your shelf made me smile

[@willwunsche6940]

Don't think I don't see that sweet Dune Lego set in the background

[@slatt9880]

Honestly one of the most informative btd6 videos I've ever watched keep it up

[@Bone_Incidents]

Taught me better than school ever did.

[@Diego-or1kv]

You forgot to take into consideration the monkey knowlege that makes valuable bananas better

[@Schloempel]

Maybe the best Banana Farm was the friends we made along the way

[@flywingcpy7121]

This is a very cool video! Unfortunately, I started getting lost near the end because my math knowledge couldn't catch up anymore. It was still cool to see the methods i used in Economics be applied to a game i love dearly. Very glad you discussed how taking out the banks deposit to reinvest is better than leaving it to compound interest! I thought i was so smart when i figured that out.

[@Т1000-м1и]

I have a feeling that Asian schoolchildren can't force themselves out of those thinking patterns and must keep doing maths even while playing games

[@ashie.official]

this is genuinely such an interesting video to me because i've recently been doing my own work on stardew valley crops-- writing a calculator to determine how many seeds of each crop you should buy under whatever circumstances. i had to do a lot of work in determining and testing my own metric, since gold/day as the wiki suggests is misleading, for very similar reasons you presented at the start of the video (e.g. cheap parsnips vs expensive starfruit). glad to see someone else going to the same (slightly crazy) lengths as me to check the math on this stuff :)

[@ashie.official]

more details for anyone curious: there are a certain number of finite resources, namely money, energy, space, and time. how do you maximize your profit? well, maximizing profit isn't quite right, since getting 100 gold in 1 day is better than 200 gold in 10 days. so let's optimize gold/day? okay, but even if an expensive crop has better gold per day, you might be able to get better profits by planting more of a cheaper crop (same as what was mentioned at the start of the video). then we also have to deal with the fact that seed price doesn't matter as much when we have a large budget. but we can't just multiply the metric by our budget, since it just multiplies every seed's metric equally. well... i eventually arrive at log(budget/seed cost) * profit/day for my metric. from there, it becomes a linear optimization problem. our constraints are energy, budget, and space, and our objective function is just a linear combination of all crops' metrics. it was such a fun problem to think about

[@alexonscience]

Interesting! For a minute I thought you were talking about agriculture with real crops lol. Would love to see it if you made a video on it too!

[@shatemmo]

Delta was incorrect for most of the banana farms because banana farms produce multiple bananas per round, so for example delta for the base banana farm would be 0.25 since it produces 4 bananas per round.

[@joshhall9436]

I love how this feels like a school project and a 3b1b video at the same time

[@datgio4951]

My guy I just want to say YOUARE CULTURED for having that Ornithopter in the background

[@randallrogers4270]

Great video! Id love to see this knowledge applied to a specific game example, showing what the most efficient purchasing route would be in real time

[@requispwarship]

this unironically helped me understand these concepts better than any lecture

[@sythrus]

I love when games create the circumstances for absurd math scenarios to exist

[@dillonconners586]

Best tower is the forest druid. Stops everything but larger MOABs by itself AND produces more income per round than most banana farms. If you can unlock it early enough, it pays for itself many times over, while having the greatest amount of pops on your board.

[@Mycatsveaissocute]

Math is fun when it´t optional, aka this.

[@Lukeepoo69]

It is 4:30 AM and I am watching this instead of sleeping