Today we have
Archimedes' cattle problem. It was, for a time,
probably the most famous unsolved problem in maths, but now no one has really ever heard of it,
but it's a great story and I'm gonna tell it to you Brady.
(Brady: I'm ready for it!)
Maybe do we need to know who Archimedes is?
Yeah, I think everyone knows who Archimedes is;
like the most best, brilliant, most brilliant of the ancient mathematicians.
Anyway, this
problem was not known about
until in 1773, the German intellectual, he was a playwright, uhm
this famous intellectual, Gotthold Ephraim Lessing,
who at that time was the librarian at Wolfenbüttel, which was the great North European library,
and discovered a manuscript written in Greek that no one had ever seen or picked up or commentat-
commented on before. In this manuscript, it said it was a letter from
Archimedes to Eratosthenes who was his contemporary, who was actually at the Library of Alexandria.
And it was a poem,
rift of twenty-two rhyming couplets, in Greek,
which said this problem; and I'm gonna - my ancient Greek isn't very good
so I'm going to transcribe the problem into the language of today.
So the Sun-God -
would be quite nice if you'd do a graphic of the Sun-God - had a herd of cattle, that's why it's called the cattle problem.
(Brady: With all that yellow you're wearing. You're the Sun-God!)
I am the Sun-God.
I had a herd of cattle and Archimedes, from Syracuse,
see like ooh he's there, in
Sicily. So the Sun-God had cattle and the cattle was all on the isle - island of Sicily.
There are bulls and there are cows and they come in different colours and this is
how we describe them:
the white bulls are a half plus a third
of the black bulls, plus yellow bulls.
The black bulls are:
a quarter plus a fifth of the dappled bulls -
now, my first comment here is that this is an absolutely terrible problem. It is so boring!
It's incredibly dull and you're gonna have to speed it up.
But actually, do carry on because we do get to why is actually really quite fascinating.
[mooing]
White bulls
plus
black bulls
equals
square number.
And the dappled bulls
plus
yellow bulls
equals a
triangular number.
And we have it. So just to point out a few things here that might be interesting.
We've got bulls: male, cows: female.
We have only got unit fractions.
Our square number,
so that's a number that's a square of another number.
So, 1, 4, 9, 16, 25, etc.
A triangular number. You must have done many things with triangle numbers, but they are the ones that you make triangle of,
so that's 1, and then 3, and then
6, and then 10, basically when you do dots together that makes a triangle.
And herd is,
cows and bulls together: both genders.
So you have this, I'm looking at it, and I get this cool dizzying effect that this is
the world's most horrible
simultaneous equation question.
What we would do is that we would write it all out,
You know call white bulls X, black bulls Y,
uhm, dappled bulls something else.
With their nine equations in however many unknowns and then we would start to work it out.
So,
what happened? Back to the story. When Lessing in 1773 found it
he wasn't a mathematician, and he went to his local mathematician and said 'can you solve this?'
And, it's true. What you do is-
let's go - ;cause the first section is really straightforward.
This is something that we could probably do. Many people really could probably do.
You would get an answer of:
it's just over fifty million cattle.
The herd has about 50 million.
Which means, on Sicily, 'cause I'm the Sun-King and I want to know how many makes my herd,
you've got one cattle,
one piece of cattle, one animal for every about 500 square meters.
Okay? But this is a problem written as a poem and in this kind of commentary on the problem as we go, and he says:
Uhhm,
don't get smug and say that. It says:
"Even if you've got this far, thou cannot be regarded of high skill."
So there's some idea that the person that's setting this puzzle is kind of
taunting you.
Let's then try and include the white bulls plus the black bulls is a square number.
This is something that was also solved shortly after it was discovered. And now the solution is:
51 trillion.
So the number in the herd is 51 trillion.
In other words, if you've tried to put all these cattle on the Isle of Sicily.
Sicily is completely covered in cattle.
You can't see one square centi-, we'll go milimeter of Sicily, and the cattle are stacked
dozens if not hundreds of meters high.
So it's this massive mound of groaning, mooing cattle.
Now on to the ninth line, because now is where things get really difficult.
In fact, if you include the line dappled bulls plus yellow bulls is a triangular number,
this is now beyond the realms of 18th century mathematics.
No one can solve it, and it becomes a celebrated problem.
It gets around. This is the Archimedes problem! We're so close. So close to solving it!
But no one can do it. At the beginning of the 19th century
Gauss, the great German mathematician, the best mathematician of his age,
it's rumored that he has a proof but no one ever sees it.
And it's only a hundred years after the puzzle was first discovered that the first person
kind of gets the answer.
And this is August Amthor, a German, another German mathematician, and he doesn't get the full answer.
He just realises that the answer
begins with the digits '766' and continues
for another 206,542 digits.
Okay, so, how big is this number? This number in terms of the herd -
I'm the Sun King on Sicily -
you're gonna start layering Sicily, put the bottom layer of cattle: dappled, yellow, whatever.
Start building them up. You're gonna go, you're gonna go out the atmosphere. You're gonna then go beyond the moon.
You're actually going to go beyond the Solar System, out the galaxy, and go to the end of The Universe and just -
only just beginning. It's gonna blast through the end of The Universe.
In fact,
this number is so big that if every cattle was an atom,
there's not enough atoms in The Universe as we know it.
It is such an enormous bonkers number beyond all comprehension
Okay? So, why would we ever want to know this number?
Well, some people did. So a couple of years after
three friends in America decided they wanted to tackle this problem and they spent four years on this problem.
They only got another 44 digits. Imagine that!
So this is something that people wanted to know. It was this kind of, kind of crazy Archimedes number.
It was only in 1965, with the advent of super computers, that the full number was actually drawn and printed out in its entirety,
and it took about seven hours
and stretched to 42 sheets of A4.
So,
What can we say about the authorship of this problem?
Lot's of people have said 'how do you know it's Archimedes?'
Well...
we don't know. All we have is the problem and it states that it's from Archimedes to Eratosthenes.
Some academics, some researchers, think 'yeah, who knows? It probably isn't.'
But a lot of them do!
And one reason why it might, because we know that Archimedes is really, really interested in massive numbers.
In fact, in The Sand Reckoner, one of his most famous kind of, treatises is
he invents a whole system of numbers to try and describe,
try and count, the number of grains of sand there are in The Universe.
So we already know,
he's interested in this area and maybe this problem wasn't something to be solved,
but just a demonstration.
Kind of the, awesomeness of mathematics to show that with
nine very simple statements, statements using nothing complicated,
you can actually create a problem that describes a number
utterly beyond comprehension, and that is going to take 2,000 years to solve.
I mean surely, that's some kind of genius.
If you wish to please the Sun-God - well, why not check out his latest book?
"So You Think You've Got Problems?" it's by Alex Bellos, it's full of puzzles and teasers like the one we just discussed.
It's the latest of Alex's mathematical books and I'll include some links in the video description,
along with more of our videos here on Numberphile that we've made with Alex over the years.
Thanks for watching.