The Archimedes Number - Numberphile

Archimedes's famous Cattle Problem had a truly epic solution. Featuring Alex Bellos.
More links & stuff in full description below ↓↓↓
Alex Bellos: www.alexbellos.com
More Alex videos on Numberphile: bit.ly/Bellos_Playlist
So You Think You've Got Problems by Alex Bellos
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The Cattle Problem by AH Bell: www.jstor.org/stable/pdf/2968...
The Cattle Problem on Wikipedia: en.wikipedia.org/wiki/Archime...
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[@goatgod2009]

"Went to his local mathematician." I want a friendly neighborhood mathematician... 😥

[@zjvuiunkgzkgvb]

surely he is referring to some academy person, which is available for consultation even today :)

[@skeletonrowdie1768]

Friendly sure is an assumption 😜

[@spirk3144]

Me too! 😖

[@littleratblue]

As a kid, I sent an email to Wolfram, asking if I had discovered something. About a decade later, they emailed back, "Nope". But, I do suspect that there's a math teacher somewhere in your neck of the woods.

[@corpsiecorpsie_the_original]

@Suki Desu - Dustin's comment was a joke

[@venisontron]

It's a good thing Lessing Gotthold of that manuscript

[@jamdc2000]

I love this community lol

[@mebamme]

Dang, you beat me to it! :)

[@idndyzgaming]

Out. 😂

[@hps362]

I legit didn't realise they meant his name was Gotthold until they said he found the letter. I thought the sentence was finishing "Gotthold of the letter" or words to that effect.

[@santerisatama5409]

Anybody else conceive 'Lessing' as a verb? To rephrase Bucky, "By Lessing we can Moring".

[@Sirenhound]

At some point the cattle are no longer standing on Earth, Earth is standing on them.

[@snowgods2195]

The cattle easily have sufficient mass to collapse into a black hole.

[@alansmithee419]

@@snowgods2195 pedantry warning: So do protons, but yeah I get what you mean.

[@zzzyzzzyzzzyxxx]

That is a solution to another problem; it isn't turtles all the way down, it's cows.

[@X_Baron]

Not cows but "pieces of cattle"!

[@neilgerace355]

@@X_Baron The only non-gendered word I can think of is oxen

[@alansmithee419]

6:55 I have rarely herd such an understatement. Number of atoms in universe: ~10^80 Number of cattle: ~10^206000 They're incomparable.

[@alansmithee419]

@@epsi you mean imaginary numbers? Yeah they're fun.

[@dlevi67]

@@epsi Get real!

[@tim40gabby25]

If each cow is an atom, fill the Universe, shrink to the size of an atom, repeat twice.. 3 of these should do it.

[@alansmithee419]

@@tim40gabby25 so let's do that. We've got a universe containing 10^80 particles, each of which contains 10^80 of the below universes' particles. That's (10^80)^2=10^160. Now each of those 10^160 particles contains 10^80 giving 10^160*10^80=10^240. Still incomparable. You don't cube the exponent (80^3 = 512000 giving 10^512000), you cube the whole number ((10^80)^3=10^240). You'd have to do this 2575 times to get (10^80)^2575=10^206000=~the number of cattle

[@RedwoodRhiadra]

What andrew meant is not a universe of 10^80 atoms, but a universe of 93 billion light-years diameter, packed solid with atoms (as he said, "fill the universe"). So a much larger number of atoms in each notional universe.

[@Hecatonicosachoron]

What I would like to know is how Archimedes proved that a solution exists...

[@stevethecatcouch6532]

That would be more interesting than the solution. The technique used to solve the problem with the last two requirements involved Pell equations. Pell equations are reported as having been first studied by Brahmagupta 400 years after Archimedes lived. It is unlikely that Archimedes solved the problem, but he might have determined that a solution existed.

[@MusicBent]

Firstly you have a solvable system of equations. Enough formulas compared to variables that a solution should exist. For example: x=1 - 1 equation, 1 variable. x=y, y=2 - 2 and 2 ... This isn’t a proof, but the system of equations only uses rational numbers. There is an infinite number of these (far fewer than the real numbers). There is no reason for a solution not to exist.

[@antonyzhilin]

@@MusicBent essentially, this is an underdetermined system, in integers. Some of them are unsolvable, so 🤷‍♂️

[@travisrhodus1362]

Yeah if it's solvable by rational #'s, then it stands to reason that there should be infinite solutions. Dat triangular kicker tho..

[@pyglik2296]

@@MusicBent Yes, but we've got eight variables and seven "normal" equations, which gives infinitely many solutions and then two extra, that are different. They point out that variables are in certain sets. If from the first we know that solutions are in form of x=q*x_0, where q is a rational vector, we don't really know if there is any x_0 for which both (or only one of them) of the specified x'es belongs to those sets.

[@SimonTiger]

6:47 Well, that's kind of an understatement... For this number, you could replace every atom with _another whole universe,_ and replace each of those atoms with new universes, and so on, where you can keep going _more than 2500 times,_ and still not have enough atoms to make the value of that number.

[@Pining_for_the_fjords]

There are two cows in a field. One says "MOOOOOOOO". The other says "Damn I was about to say that!"

[@nate0___]

4 years later, and I could see this in an asdfmovie.

[@edderiofer]

I wonder if the original problem might have had a much smaller solution (one that Archimedes and Eratosthenes would have been able to compute), and if, somewhere along the line, it had been mistranscribed, causing it to have a much larger solution than it was originally intended to have?

[@PatrickRyan147]

Ans: 42.. sheets of A4 paper to print the answer on..

[@Shadow81989]

Why so low? I had to use the search function to find this ONE comment about 42 being (included in) the answer!

[@sakurahikxri]

Patrick Ryan too low

[@SilverLining1]

"Alright guys, here's a little exercise for fun." *takes 2 thousand years to solve*

[@Lexivor]

Imagine if instead of a herd of cattle the sun god had a forest of TREEs.

[@Wemdiculous]

Ok, but thats COWS(1) and its way bigger than TREE(1)

[@Lexivor]

@@Wemdiculous What would COWS(2) be?

[@neilgerace355]

Can't see the forest for them

[@Shadow81989]

@@Lexivor COWS(2) is the second smallest number that solves the equation, I guess.

[@Lexivor]

@@Shadow81989 That makes sense. I think the second smallest solution is roughly the size of the square of the first solution.

[@yoloswaggins2161]

It's amazing they got the sun god to do a video for them. He looks quite a lot like Alex Bellos but maybe a bit younger and more handsome.

[@BainesMkII]

This leaves out how people were gradually able to calculate individual digits of the answer while being seemingly incapable of calculating the full number.

[@fantafanta2823]

That was the most baffling part to me! Where's the Extras video?

[@ceruchi2084]

I agree with you, but I also have a simple guess. I think it's got to do with the fact that each digit is another order of magnitude. It's way easier to say when a bowl of rice has between 10,000 and 100,000 grains than to say when it has between 1,000 and 10,000 grains. This is still true on the most massive scale.

[@ananyasharma6239]

5:45 oh these rumours are dangerous to hear. Just like Fermat leaving a note that he had a proof for Fermat's last theorem.

[@kshitijsharma759]

Yes

[@archanamotagi1675]

"They say Gauss had a proof but no one saw it" -Top ten Anime endings

[@BubbleManxx]

Dubious existence of a proof. Sounds familiar....

[@naphackDT]

Probably started with the first and last sentence of the problem and worked his way from there.

[@Triantalex]

false.

[@neilgerace355]

0:28 "You said intellectual twice." "I like intellectuals." (Blazing Saddles reference)

[@scottmuck]

Eh, if you can print out the number Ron Graham is not impressed.

[@piguy314159]

Even Matt Parker and Curtis Cooper wouldn't be impressed

[@tomiesz]

They printed it though

[@haronka]

But TREE(3) is still Way bigger

[@mmeister8582]

tomiesz there are not enough atoms to print it

[@scottmuck]

tomiesz exactly!

[@emberke100ful]

There is more cattle in Sicily, than there are atoms in the observable universe! - Some Greek Guy

[@MaxKuschmierz]

9 lines and 2000 years to solve? Laughs in collatz

[@dlevi67]

Collatzes into laughter?

[@feedswav]

collaughz

[@santerisatama5409]

Or what do I know... maybe Archimedes Number is the first that computationally brakes the Collatz!!!?

[@Triantalex]

??

[@madcapprof]

How the heck did Archimedes know that the complete problem even had a solution?

[@bokkenka]

Who says he did? Maybe he was just poking at Eratosthenes. "Here... If you can't solve this, you must be an idiot."

[@Jodabomb24]

It's an underdetermined system, so it actually should have an infinite number of solutions.

[@Lexivor]

@@Jodabomb24 It's kind of amazing to think that the 206,545 digit solution is the smallest of an infinite number of solutions.

[@merdehappens]

@@Lexivor Only if you ignore the last two statements.

[@Zaros262]

Joseph McGowan it's not obvious that there are an infinite number of solutions that are square and triangular, and if there are an infinite number of solutions, that would be an interesting proof There are underdetermined systems that don't have any solutions at all, so your assumption doesn't hold up. Edit: removed bad example Edit: better example of a simple underdetermined system with no solutions: {x=0, x=1, a+b+c+x=2} Four variables, only three equations, but still 0 solutions

[@p11111]

8:39 "udderly" 😳

[@lemmysverruca]

Oh a cow pun. I've rarely herd these before.

[@p11111]

lemmysfibroma Do you have a beef with my comment?

[@nadgerz]

@@p11111 - Perhaps @lemmysverruca's comment is moot?

[@janus3042]

Me after seeing the Problem: Well, I would say there isn't a single cow on Sicily; Problem Solved

[@egilsandnes9637]

Brilliant!

[@CaptainBohnenbrot]

You ... you ... are right, sir.

[@Hecatonicosachoron]

Yeah, the solution would be that Odysseus’ troup killed them all

[@genessab]

Nr.3042 but 0 isn’t a triangular number so it doesn’t work.

[@CaptainBohnenbrot]

@@genessab 0 is regarded as a triangular number by most people.

[@JDescafeinado]

Please, could you make another video explaining how exactly a mathematical calculation can lead you to know the first 3 digits of your solution, and also its exact lenght? Minute 5:50

[@Lexivor]

In short, Amthor figured out that the solution involved raising a more than 20 digit number to the power of a four digit number. This was too long to calculate by hand, but using logarithm tables he was able to get an approximate solution.

[@Bigfoot_With_Internet_Access]

Just flexing on everyone who can't even count

[@travisrhodus1362]

Archimedes 2K years later "I've earned this $50 joint 😁"

[@deminidze]

ima flexing on those who couldnt read that in ancient greek!

[@mahbubmaleksyed5301]

Hello again

[@Triantalex]

??

[@Flatunello]

Archimedes is my hero. As an engineer, even at age 66, I hold this sage in the highest regard. To the engineers I hired for my firm in California, I always had them write an essay on an aspect of his writings they found interesting and demonstrate the mathematics. It was a quick way to discover who loved engineering and who didn't.

[@AstAMoore]

This really puts the cows in the expression “spherical cows in a vacuum.”

[@Nilguiri]

Cows are a torus as any fule kno.

[@CaTastrophy427]

I love turning anything into something described by spherical cows in a vacuum. I once set up an english literary analysis project for a 10th grade project that seemed normal, just with the occasional slight tangent, until I got to the end, where I spent a single paragraph to show that, given my normal analyses of the individual portions, the argument posed by the book could be applied equally to spherical cows in a vacuum. Then, I logically restated it to fit those cows in a less confusing way while still including every major and minor point of the original argument. And then, once reframed to do so, I showed that it could then be reframed back to human societies, in a way where the transformation makes logical sense, but the end result is diametrically opposed to the original argument. As such, I showed that the book's takeaway, the moral of the story, was utterly (udderly) meaningless. Oh, and this was an oral presentation in front of the entire class. The teacher tried to stop me when I started talking about spherical cows in a vacuum, but I said "This is the last paragraph, and it's relevant. In fact, it's the entire point of my project" before restarting the paragraph. I knew that it was somewhat confusing, so I came prepared with about two dozen printed copies of my speech (each one was just one piece of paper, double sided) to hand to anyone who wanted to read it over. I got a begrudging A+ on the project. TL;DR: Showed a persuasive argument to be meaningless by reframing it to spherical cows in a vacuum and then reframing it back to its original subject.

[@justahker3988]

@@Nilguiri Only when they open their mouths.

[@ShDragon1]

@@CaTastrophy427 Please tell me you still have copies of that saved somewhere!

[@carbrickscity]

They probably want to be careful with the phrase "big number" after they covered Graham's number and TREE(3).

[@RedwoodRhiadra]

Not to mention TREE(Graham's number)...

[@carbrickscity]

@@RedwoodRhiadra And Googolplex for instance.

[@carbrickscity]

They did Rayo's number though lol

[@hexagonist23]

@@carbrickscity tree(g(rayo's number))

[@omerd602]

"Bigger than the number of atoms in the universe" is an understatement. (I'll be referring to the observable universe when I say universe.) If you were to add an atom into a new universe (universe 2) every time you counted the number of atoms in our universe (universe 1), and then added an atom to universe 3 whenever universe 2 filled up, and keep doing that for as many universes as you need, you would still need several thousand universes to count to this number. This does not mean the number of atoms in several thousand universes, just to be clear; it means that, to quote another person who used this reasoning, "We're writing it in base universe."

[@oskarekberg3704]

Aaah... the Cattle problem. I've herd about that one.

[@LPNurja]

Haha, I was actually born and raised in Wolfenbüttel. It's called the "Lessingstadt" (Lessing city). The other thing the city might be known for is Jägermeister. ;) We don't have that many cows though...

[@leadnitrate2194]

Oh I've heard of this problem. Always wanted to see it on Numberphile, because of the large numbers involved.

[@adrianogiacomini6541]

Isn't 0 a solution?

[@nathanderhake839]

Adriano Giacomini the FBI wants to know your location

[@DanielQRT]

@@adrianogiacomini6541 yes but it's a trivial solution

[@hehmann3215]

Holy number that is a big cow

[@capitalist88]

I did what you see there.

[@Triantalex]

??

[@broyojo]

Who else though Gotthold Ephraim Lessing looked like a cone head?

[@maxamedaxmedn6380]

Exactly

[@toweypat]

He comes from France.

[@vikraal6974]

He looked like Ice Cube

[@frogfan449]

he got a 5 head :/

[@Nilguiri]

Haha, that was the first thought I had about him!

[@woodyblox7724]

5:58 hätte nicht gedacht, dass ich den Namen „Amthor“ im Zusammenhang mit Erfolg hören würde

[@Rugged-Mongol]

Wir sind nicht so dumm.

[@Irondragon1945]

HA

[@liliwheeler2204]

The answer to life, the universe, and the number of pages filled by the digits of this cattle number

[@dozenazer1811]

1:00 Was it made in 2012?!

[@crimsoncanvas51]

Farmer to Archimedes : I want my cattle to be remembered in history . How can you help me? Archimdes : I need to create a problem for this problem then .....

[@anarcho.pacifist]

The solution can be computed by solving the Pell equation: x^2 - D*y^2 = 1, where D = 410286423278424. The smallest such solution to Pell's equation is: x = 37653445023472058840...(103233 more digits here)...84777023371728320049 y = 18589219013869133094...(103226 more digits here)...92592107710208663490 From this, we find: k = 4456749 * y^2 B+W = 17826996 * k (this value is a square number) D+Y = 11507447 * k (this value is a triangular number) Where the total number of cattle is: B + W + D + Y + 21054639*k = 77602714064868182695...(206505 more digits here)...23406626719455081800 See also OEIS A096151.

[@maxmouse3]

I love these classic numberphile videos ♥️

[@regulus2033]

I wonder, how did that man calculate FIRST 3 digits of this number? Usually it's easier to find last several digits (by modular arithmetic), but first... Interesting.

[@aliedperez]

_"and he went to his local mathematician"_ I wish I had a local mathematician I could go to :)

[@jakelucena3392]

“this is a bonkers off the scale number” graham’s number: hold my beer.

[@Solar424]

TREE(3): Amateurs

[@CrystalblueMage]

42! Sheets... A-HA!!!

[@questionable-cf1tt]

My dudes, after Tree 3 and Graham's number and now this it would be interesting to know some of the smallest numbers ever used in math problems 😄

[@tochoXK3]

negative Grahams number? Or zero, if you ignore negative numbers

[@inigo8740]

Archimedes certainly thought highly of Sicily and their farmers.

[@hienduongvan8454]

0:33 i reckon thats a Steel Cap hes wearing (Weight: 8, Defense: 8-12)

[@literallylegendary6594]

"German intellectual, playwright, famous intellectual" *Ah yes, the floor is made of floor*

[@Martin-qb2mw]

Best video in a while. Great stuff!

[@isaak.studio]

Gotthold Ephraim Lessing was quite an egghead.

[@BleuSquid]

An alternate solution for the problem as presented in the video is 0. The full problem also states "The number of bulls is greater than the number of cows" which precludes 0 from being a solution.

[@denelson83]

8:39 - Not "udderly behond comprehension"?

[@Neko91485]

“It’s a stupidly big number” Graham “hold my calculator”

[@danielm.1441]

Archimedes' Cattle Problem? That's how he Gotthold of it.

[@yamomwasthebomb7159]

The written descriptions at the beginning reminded me of Drunk History. And now I want a Drunk Math.

[@roderik1990]

So I got an interesting visual illusion at 7:35 when the different pages full of numbers showed up. Some of the numbers seemed to shift after the fact. Did something along the line of accidental alignments of numbers messing things up? Did they actually change after the fact? Or is this just some weird visual glitch?

[@edghe119]

Holy Cow that’s a big number. But looking back, tree(G#) makes this number seem comprehensible.

[@fernbedek6302]

Add this to my list of things to do with a time machine: give Archimedes the full print out.

[@kevingil1817]

It took 7 hours to print 42 sheets? This has to be a very holy number!

[@BobOgden1]

Finally! Sufficient cowbell

[@seamusriley3532]

Underrated

[@samsungsamsung-hd3bj]

Nice video. Could I ask what is OST in the end of the video video ( should be chorus) ?

[@VidMashUp]

Lol! You said "utterly beyond comprehension"!

[@ebtsoby]

that's crazy, took over 2000 years and the invention of the supercomputer to solve his question, truly a genius

[@CrashingThunder]

It makes me feel lucky to live now, in a time where if we want an answer to a fairly straightforward problem, all it takes is a few dozen lines of code to crank it out. We can literally just tell a machine how to find an answer, and receive it not much later. What's even cooler though, is the realization that there are problems today that even our supercomputers cannot solve, but might be trivial in a century or even just a few decades.

[@SuviTuuliAllan]

In contrast: It only took 15 years and three supercomputers to MacGyver a DHD for the Tau'ri gate.

[@bestproductable]

It doesn't really take a genius to make puzzles, he didn't find the solution himself either

[@willcrapenter1779]

Numberphile viewer: See’s Title Numberphile viewer: Screams in TREE( Graham’s Number )

[@haronka]

I wonder whether they will make a vid about Loader's number

[@erik-ic3tp]

@@haronka, That would be cool too. :)

[@soranuareane]

I'd love to see the system of equations used to solve this.

[@Tevildo]

Kaedenn - The initial equations (Archimedes' "no expert" level) are as follows. Unknowns are Bw, Bb, Bd, By, Cw, Cb, Cd and Cy. Bw = 5Bb/6 + By Bb = 9Bd/20 + By Bd = 13Bw/42 + By Cw = 7(Bb + Cb)/12 Cb = 9(Bd + Cd)/20 Cd = 11(By + Cy)/30 Cy = 13(Bw + Cw)/42 Note that we have seven equations and eight unknowns, so the solution to the first part is not unique; this makes the second part solvable.

[@Galakyllz]

This was excellent. Thanks for the video.

[@numberphile]

Glad you liked it.

[@kevin-y]

Did Brady get a new editor? Loving the captions and little animations on this one.

[@DurianFruit]

The captions are written in invisible ink

[@Marconius6]

Of course knowing ancient Greek philosophers, the real answer is... "There are no cattle in Sicily."

[@Todd_Manus]

Ummm.. I think we have an issue... @6:12 you have the first three digits as 766.. but later in the video you have several references that state 776 as the first three digits. Is that a mistake?

[@CB27255]

I wish we could see more actual mathematics in these videos. Perhaps ideas or small snippets of logic from the proof. I feel like we didn’t get any insight into how the problem was solved or why it was so hard etc. Love the channel though :)

[@Kedzke]

But if you add an extra equation to the system, you don't get a new answer, but just reduce the set of possible answers. Therefore that 766... number should match the original system as well (i.e. without square and triangle number conditions). So perhaps the problem is to find the *smallest* number of cows.

[@TheChefCain]

Yes of course, the most best brilliant most brilliant of the ancient mathematicians, how could I forget?

[@yoyamon6811]

Sometimes I feel like mathematician create and solve these problems just to flex xd

[@Lexivor]

You are correct.

[@owen9643]

Is it right that the number of atoms from the largest number would be more than in the universe because if you divide 206542 by 23 which is the power in Avogadro’s number you get around 8980.087 so it would be somewhere around that many mols of an element and if you use hydrogen and find the mass of that you get around 9051.928 grams of hydrogen which isn’t close to the mass of a Star with lots of hydrogen in it

[@Gooberpatrol66]

Me: can't invent math problems or write poetry Archimedes: writes math problems in the forms of poems

[@ahmedalnazer5810]

Imagine going to your local mathematician 3:45

[@billpg]

At the time, was it the largest finite number to be precisely defined?

[@pacman52280]

Where would I find the complete number if I wanted to print it out for myself?

[@murk1e]

Those captions for Archimedes and Lessing feel like a little friendly dig from Brady :)

[@jivejunior8753]

Why did they change the title from "Hoy Cow, That's a Big Number"?

[@jcughan]

Udderly beyond comprehension I’ll see myself out...

[@benjiboi69_vlogz87]

Didnt this used to be called "Holy Cow Thats a big number-Numberphile"

[@omikronweapon]

Stuff like this makes me want a timemachine. I'd like to see Archimedes come up with this. Why? How? Was it just for a laugh? What did he expect to learn from it? Did he expect others to solve it? Was there any reasoning behind the fractions chosen? Seeing he chose Sicily as the area, did he expect it to fit in any way? Could he even imagine the actual scope of the answer?

[@sid025]

Loved it 😍

[@glenplonk]

Archimedes : Hey, here is a very big number ! Matt Parker : Hold this three-book-long printed prime number !

[@saltech3444]

Does anyone know where the Cattle Number is online? Just so I can look at it? If it's only 44 pages long, it should be on some website somewhere!

[@kabinpathak9233]

I have been working on research of prime number since 3 years amd i believe that i can list prime number continuously( all) using a two alzebrical formula Is that a progress in prime number or not?

[@SaryTheWolf]

Artemis: Brother, we need to talk about your herd hording problem.

[@KedarOthort]

I don't know why I suddenly realized it just now, but I just realized they used that shot of Sicily from space for the Isle of Selini in Babylon 5, just mirrored.

[@UrnestHemingouey]

That seems like ancient trolling, trying to waste a lifetime of a friend

[@kshitijsharma759]

Lol 🤣

[@additionaddict5524]

I wonder if there’s a simpler version of the problem in order to get an understanding of why this blows up

[@superlukey3]

This may not apply precisely but this will hopefully help. One problem I worked on a while ago, from Project Euler (lots of cool math problems that have big numbers) was to find the lowest number which was a multiple of all integers from 1 to 20. That can be thought of as a simple version of this problem - a handful of equations, some constraints, what's the lowest possible number. Well, let's say we only do one constraint - that the number must be a multiple of 2 (we can drop the multiple-of-1 argument, that is trivially true). That's easy! Just look at the even numbers, the lowest answer is 2. So now, add one more constraint - it must be a multiple of 2, and a multiple of 3. This is equivalent to saying it's a multiple of 6, by the way - thinking back now I could have made the problem easier if I'd thought about that, but anyway. Suddenly the numbers just got a lot more spaced out. Now it's no longer every other number, but only one in every six numbers meets every requirement. Now add that it must be a multiple of 4 as well - that makes the lowest number 12, only one in twelve numbers meets the requirement. And, a number that is a multiple of 2, 3, 4, and 5 (and 6 by default)? The lowest one is 60 - and 60 isn't a multiple of 7, 8, 9, 11, 13, 14, 16, 17, 18, or 19, so better keep looking. Every additional constraint weeds out a ton more numbers and pushes back the first number that satisfies the equation a ton. I don't remember the final answer to this problem, but I do know I wrote a computer program to solve it rather than even attempt to do so by hand, and it was a large number. I hope this helps makes it clear how adding constraints like this leads to large numbers, fast.

[@additionaddict5524]

@@superlukey3 Answer will be (2^4)(3^2)(5)(7)(11)(13)(17)(19). Think about prime factorisation of each number 1 to 20.

[@superlukey3]

@@additionaddict5524 Ooh, clever! Ultimately my program started at 20, jumped forward 20 numbers at a time, and just checked the modulus of each number in reverse order (so checked for 19, then 18, then 17... down to 2). My immediate thought was just cutting down on ones that wouldn't need to be checked, ie multiples of 18 are multiples of 2, 3, and 6 as well so no need to check those again. Even so, my naive approach solved the problem pretty fast - but your analytic solution is much better!

[@awesomeawe]

Love Alex!

[@courtney-ray]

0:59 AOL is definitely just about as ancient as this problem is! 😄

[@Arkantos117]

All these mathematicians have been bamboozled, they had no yellow cows!

[@philipcollier4883]

Ok this problem reminds me of the 17 horse problem: A farmer dies and leaves 17 horses to his 3 sons. The will stipulates the oldest son gets 1/2 the horses, the second eldest 1/3 of the horses, and the youngest 1/9. An old man walks by and hears the sons arguing about how to cut up the horses and offers them one of his old mares. With 18 horses 1/2=9 1/3=6 1/9=2 for 17 horses. Having resloved the dispute the old man took his mare back and continued on his way. My question is did anyone try adding "virtual cows" to this equasion to make the numbers more reasonable?

[@hammerth1421]

Why does Numberphile remind me to do my homework? I never wanted to hear 'Gotthold Ephraim Lessing' ever again!

[@lemerdtool]

I had both H.C. Williams and Zarnke as professors, did not know they worked on this problem.

[@Number_Cruncher]

I cannot reproduce the intermediate result 51,285,802,909,803 pieces of cattle. The solution of the first 7 equations gives for the total sum 50,389,082 * k. And with the next condition, at least k = 3*11*29*4657, which gives a total sum of 224,571,490,814,418. What do I miss?

[@deidara_8598]

0:28 You can tell from the shape of his head he's been binge watching Rick and Morty

[@pierreabbat6157]

I don't blame you for not being up to reading the Greek. It's a bit more archaic than the Greek I'm used to, with a genitive in -οιο in the first line, and there are several words I don't know. The dappled cows are referred to as μικτοχροοι, ποικιλοι, ποικιλοχρωτες, and maybe other words.

[@andipandi5641]

i still want to know more specifically how many there were of each cow or bull ??