Why Archimedes is probably the coolest Mathematician of
Why Archimedes is (probably) the coolest Mathematician of all time -And the problem that proves it -Inspired by ‘Archimedes Revenge’ by Paul Hoffman
So you think you know Archimedes? Here are some of the things he did in his life The Archimedes Screw The Archimedes Principle of displacement Doing a lot of maths about the equilibrium of planes and helping to exploit the use of mechanical leverage He was also the subject of legend, and was attributed many feats that simply weren’t true He did not: • Build a giant claw to defeat the roman siege by tipping ships over • Build a laser beam using mirrors to burn attacking ships • Run naked through the street dripping wet from a bath screaming “Eureka” • Say “Give me a place to stand I will move the Earth” • Or refuse to leave his diagrams resulting in his murder by an invading roman soldier Improving the catapult
Archimedes' Revenge - the backstory • SO he’s a bit of a legendary figure, but today we are going to focus on a very specific question that he concocted to be his ultimate revenge! • Archimedes did a lot of work with large numbers and was one of the leading minds in the research. • Apollonius of Perga made some improvements to Archimedes working, which upset Archimedes greatly, not because it was wrong, or even because Apollonius had been disrespectful, Archimedes didn’t like being bested • He then devised a problem involving herds of cattle, taunting other mathematicians to best his challenge, the gauntlet had been thrown, and he was about to demonstrate his genius Apollonius
Archimedes' Revenge – Part I The question can be broken in two parts, the first being difficult, but not undoable, by the standards of mathematics for the time: Of all the cows in Sicily, imagine that the are all grouped into different herds based on their colour. The proportions of the herds relative to one another are represented by these equations 1. 2. 3. 4. 5. 6. 7. White bulls = yellow bulls + (1/2 +1/3) of the black bulls Black bulls = yellow bulls + (1/4 + 1/5) of the dappled bulls Dappled bulls = yellow bulls + (1/6 +1/7) white bulls White cows = (1/3 + 1/4) of the black herd Black cows = (1/4 +1/5) of the dappled herd Dappled cows = (1/5 +1/6) yellow herd Yellow cows (1/6 + 1/7) white herd You might recognise a few things 1. These are simultaneous equations, and can be written as such as I will show on the next slide 2. There aren’t any numbers in the equation, only fractions of an unknown, so what we’re really looking for (and this was the point when it was set) is the smallest possible number of cows that satisfies the equations
Archimedes' Revenge – Part I cont. So as I mentioned we can turn these words and additions of fractions into a more usable set of formulae Bulls - are represented as a capital letter of their colour (because bulls are generally bigger than cows and this is my presentation so there) Cows – are represented as a lower case letter of their colour (because I already used the capital) 1. 2. 3. 4. 5. 6. 7. W = Y + (5/6) B B = Y + (9/20) D D = Y + (13/42) W w = 7/12 (B + b) b = 9/20 (D + d) d = 11/30 (Y + y) y = 13/42 (W + w) Now that I’ve laid out the problem a little clearer, why not have a go at solving it? I would give an edible prize to anyone who submits full working and doesn’t cheat by looking at the next slide (bear in mind that this is only the first part of the problem)
Archimedes' Revenge – Enacted If anyone seriously did try that and got further than copying out the question then my respect for you has just doubled. The question is difficult for a number of reasons: 1. While giving an infinite amount of answers, because we only want the smallest, there is only one acceptable one 2. There are 7 equations for 8 unknown variables, where you would traditionally expect a 1: 1 ratio (your calculator definitely does) 3. The smallest correct answer is 50, 389, 082 which is of course ridiculous Personally I would be very pleased if I got this question right, but I would have been swiftly kicked off my pedestal by a very smug, and very dead Archimedes -”If thou canst give, O friend the number of bulls and cows in each herd, thou art not all-knowing nor unskilled in numbers but not yet to be counted among the wise” My rough translation of this, is that the worst is yet to come
Archimedes' Revenge – Part II Archimedes then added a further two constraints on the number of bulls: What is a triangular number? 8. White bulls + black bulls = a square number If a square number of particles makes a square 9. Dappled bulls + yellow bulls = a triangular number 4 9 Then a triangular number makes a triangle These two small stipulations made the entire problem untouchable for quite literally thousands of years (so it goes without saying that you should put the pen and paper down) 4 6 There is a more mathematical way of doing this, which is using the sum of consecutive integers. For example, take the second triangle (1+2+3)=6
Archimedes' Revenge – The history of the answer • In 1880 over 2000 years later, it was demonstrated that the smallest herd of cows fulfilling the 9 conditions would be 206, 545 digits long, and start with the digits 776 • That looks something like this: 77 6
Archimedes' Revenge – The history of the answer • That was of course A JOKE because there are only 20’ 000 dots in that picture so we would need 10 of those slides to really understand the colossal nature of the answer we are dealing with • In 1899 a civil engineer working in Illinois founded the Hillsboro mathematics club devoted to finding the answer. After 4 years of painful calculation they had found 12 of the rightmost digits, and 30 of the leftmost ones at either end of the whole. (although later work revealed cruelly that 2 of the digits were wrong) • Then, humanity gave up for another 100 years • In 1981 the answer was finally found by a supercomputer called Cray 1, as it was then, the most powerful computer in the world. Even then it took the computer 10 minutes to find and double check it’s answer, which was then printed on 47 print outs from the machine. • Finally, the problem was solved
Archimedes' Revenge – Final thoughts As you might gather from the length of this presentation, I might have gotten a little carried away with the story. It is therefore good to step back and realise the ludicrous nature of the whole thing • The question doesn’t even make sense, originally this number was supposed to be all the cows in Sicily which is a small 6, 358, 400 acres. The answer in part 1 could comfortably fit here, but not the part 2 answer with 206, 545 digits • It took a literal supercomputer, used for visual effects in movies, cracking secret codes, and even simulating sci-fi level space weaponry; in order to work out a problem about some cows • And 2000 years later at least 1 person (me) is still talking about it! Archimedes’ true revenge was being undisputedly the coolest mathematician of all time and truly the master of large numbers. Could you write a question that lasts over 2000 years?
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