Did Archimedes Write a Problem That Took 2,200 Years to Solve?

this math problem is about counting the number of cattle in a herd it took 2200 years to solve and the number of cattle in the final answer is so big that if you take all of the atoms in the known universe and turn each of them into their own universes then there would be more cows than all of the atoms in all of these universes combined so what was this problem and why did it take so long to solve the problem was sent to ostanes as a poem about the number of cattle owned by the sun god Helios ostanes was the chief librarian at the Library of Alexandria and the second best mathematician of his day the problem has four parts that get more and more complicated and the overall setup is more important than the specific numbers it starts like this if Thou Art diligent and wise o stranger compute the number of cattle of the sun who Once Upon a Time grazed on the fields of the thian Isle of Sicily divided into four herds of different colors one milk white another a glossy black a third yellow and the last dappled in each herd were Bulls Mighty in number according to these proportions understand stranger that the White Bulls were equal to a half and a third of the black together with the whole of the yellow while the black were equal to the fourth part of the dappled and a fifth together with once more the whole of the yellow observe further that the remaining Bowls the dappled were equal to a sixth part of the white and aeven sth together with all of the yellow so far this isn't so bad using a calculator and the modern tools of linear algebra we can solve this in seconds but even the ancient Greeks would have had ways to do this there was an elaborate geometric method that was probably in use at the time and a couple centuries later the mathematician dianus recorded a method that looks a lot like a precursor to algebra it only had a symbol for one unknown value but we could start by assigning the unknown to the White Bulls and assuming an arbitrary number for another like let's pretend there's just one yellow bll then we can use a series of substitutions to figure out the relative sizes of the other groups in the end you get these numbers but of course you can't have a fraction of a bull so we need to multiply everything by a least common denominator to get this we haven't counted the female cows yet but at the moment we're looking at about 6,300 bulls the poem continues these were the proportions of the cows the white were precisely equal to the third part and a fourth of the whole herd of the black while the black were equal to the fourth part once more of the dappled and with it a fifth part when all including the Bulls went to pasture together now the dappled in four parts were equal in number to a fifth part and a sixth of the yellow herd finally the yellow were in number equal to a sixth part and a seventh of the white herd okay this is kind of more the same for counting the female cows once more we can assign the unknown to one type of cow and substitute our way through to a solution then we multiply up the size of the entire herd to get rid of our fractions so the total comes out to 50 million 389,000 182 cattle if thou canst accurately tell o stranger the number of cattle of the sun giving separately the number of well-fed bulls and again the number of females according to each color thou wouldst not be called unskilled or ignorant of numbers but not yet shalt thou be numbered amongst the wise let's talk about who wrote this and why they thought they could challenge the great ostanes the poem was rediscovered in a German library in 1773 the inscription credits Archimedes the greatest mathematician of his time and one of the greatest who ever lived Archimedes often wrote to ostanes because not only was ostanes one of the only people who could understand Archimedes math but Archimedes would have wanted the Library of Alexandria to preserve his work for future Generations most historians agree that this problem probably really does come from Archimedes though some have suggested that the first part that we've read so far could have actually been written by aanes himself and then this next part would have been Archimedes reply but come understand also all these conditions regarding the cattle of the sun when the White Bulls mingled their number with the black they stood firm equal in depth and breadth and the plains of tra stretching far in all ways were filled with their multitude it's saying that the number of White Bulls and black Bulls added together has to be a perfect square one way that we can think about big numbers that are perfect squares is to look at their factors in order for a number to be a perfect square all of its prime factors have to be squared as well so if we take our herd of 7 million black bulls and 10 million White Bulls and we want to figure out what we need to multiply it by so that it's a perfect square we can look at all the factors like so and see what's missing we can figure out this part of the problem if we just multiply the current size of our herd by the product of these numbers so that's going to bring the total up to 224 trillion 5 71 billion 490 mil 84,4 18 cattle and we're not even done again when the yellow and the dappled Bulls were gathered into one herd they stood in such a manner that their number beginning from one grew slowly greater till it completed a triangular figure there being no Bulls of other colors in their midst nor none of them lacking if thou are able oh stranger to find out all these things and gather them together in your mind giving all the relations Thou shalt depart crowned with glory and knowing that thou Hast been a judged perfect in this species of wisdom the yellow blls and the dappled bulls have to form a triangular number here's where it goes from complicated to potentially unsolvable you'll have a triangular number of bulls if you can set them up where the first row has one bll the next row has two then three then four and so on one way to calculate triangular numbers is to recognize that I can pair off my rows when I rearrange things this way I get a rectangle where the length is equal to the original number of rows plus one and the width is equal to half the number of rows multiplying length by width gives me the total number of cows and we can simplify that a bit to get this the key to making our number of yellow and dappled bowls into a triangular number is going to be to find a value that fits everything we did before and gives us a whole number of solutions to this equation since this is a quadratic equation we can solve it using the quadratic formula we know that we'll get a whole number answer whenever this part under the square root sign what's known as the discriminant is a perfect square so now our job is to find a multiple that lets us scale up the number of dappled and yellow Bulls so that this equation Works since we don't want to mess up our perfect square from the last part this new multiple also needs to be a perfect square this gives us an example of what's known in number Theory as a pedal equation by the time the cattle problem was written Greek and Indian mathematicians had been studying this type of equation for a couple hundred years already and they'd used it to do things like approximate theare < TK of two since the equation x^2 - 2 y^2 = 1 can be rearranged like this any whole number solutions for X and Y give us a fraction that's really close to theare < TK of 2 when we estimate by ignoring that one in the numerator the bigger the pair of numbers we find the better the approximation meanwhile Archimedes himself found an impressively accurate estimate for the square OT of three while nobody knows for sure how he found it it seems likely that he used a similar strategy it goes without saying that the pel equation for this cattle problem is significantly more complicated while some think that Archimedes could have understood the setup and maybe recognized that this cattle problem did have an actual whole number solution it would have been impossible for him to have found the full value in fact it would be thousands of years before mathematicians would be able to solve this problem in the 7th Century ad the Indian mathematician Brahma Gupta had found a method for solving certain pal equations and a few centuries later bascara was able to show that this type of equation equ always has a whole number solution but even by the time the cattle problem had finally been rediscovered in 18th century Germany nobody could find an answer for many years early in the 19th century there was a rumor that CF gaus had a method for solving this problem since it was one of the most famous unsolved math problems of the time and gaus was the greatest mathematician of the era but he never published anything about it and others think he wouldn't have been that interested in a problem didn't have a wider application so it took until 1880 for Carl Ernst August amthor to work out that the solution starts with the number 776 and is [Music] 206545373 digits long about 10 years after Amor's calculation a few members of the Hillsboro mathematical Club of Hillsboro Illinois spent 4 years to find the first 31 digits and the last 12 digits finally in 1965 the complete answer was computed in 7 hours and 49 minutes on a supercomput at the University of watero in Canada even now that the complete solution's been found mathematicians have still worked on this problem in the hopes of finding simpler algorithms that computers could use to speed up these kinds of computations so why would Archimedes have sent this problem to aatosan anyway was it some kind of prank if it was it wouldn't have been out of character for Archimedes based on his writings it does seem like he enjoyed needling his fellow mathematicians who weren't quite at his level he was known to send out deliberately incorrect discoveries to see if anybody would take the bait and try to prove that something wrong was right but this cattle problem was more than just a prank it gives us a sense of Archimedes love for huge numbers something that he also wrote about in the sand reckoner where he tried to calculate the number of grains of sand in the universe in the process he invented a system for describing very large numbers and discovered the basic laws of exponents there's another theory that says that this problem was actually the final blow in a mathematical duel between Archimedes and the much younger apollonius of perga who had published several Works attempting to one up Archimedes accomplishments but if it is true that Archimedes had a sense of how to construct a solution to this cattle problem even without knowing the final answer then it would be yet another sign that a significant amount of the math that Archimedes worked out during his life has been completely lost to history okay

Channel: Ben Syversen

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Id: V2S39AgjFpQ

Length: 12min 9sec (729 seconds)

Published: Fri May 24 2024