Massachusetts Institute of Technology77 Massachusetts Avenue, Cambridge, MA, USA. MIT News | Massachusetts Institute of Technology, The answer to life, the universe, and everything, Mathematics researcher Drew Sutherland helps solve decades-old sum-of-three-cubes puzzle, with help from "The Hitchhiker's Guide to the Galaxy. The capstone senior design class in biological engineering, 20.380 (Biological Engineering Design), took on its most immediate challenge ever. Years of volumes and hundreds of essays, published by the MIT Press since 2003, are now freely available. His answer, posted in the online discussion group alt.fan.douglas-adams, was succinct: “It was a joke. For that particular problem, any solution has an absolute value lower than the square root of a given n. Moreover for the sum of squares, we know perfectly well what is possible and impossible. Adams was friends with Pink Floyd guitarist David Gilmour and came up with the title of the band's … When we do so, we see that: 03 = 0 (mod 9); 13 = 1 (mod 9); 23 = 8 = –1 (mod 9); 33 = 27 = 0 (mod 9); 43 = 64 = 1 (mod 9); 53 = (–4)3 = –64 = –1 (mod 9); 63 = (–3)3 = 0 (mod 9); 73 = (–2)3 = 1 (mod 9); 83 = (–1)3 = –1 (mod 9). Booker and Sutherland say there are 10 more numbers, from 101-1000, left to be solved, with the next number being 114. Binary representations, base thirteen, Tibetan monks are all complete nonsense. Using the Charity Engine network is also more energy-efficient. They ran a number of computations at a lower capacity to test both their code and the Charity Engine network. The numbers 41 and 43, for example, are also elements of many sequences. “He’s a world’s expert at this sort of thing,” Booker says. For integers less than 100, just three enigmas remained: 33, 42 and 74. Note that for some integer values of n, the equation n = a3 + b3 + c3 has no solution. “You wait and wait and just when you’re at the point of giving up, the number shows up,” said Sutherland. Understanding antibodies to avoid pandemics, An intro to the fast-paced world of artificial intelligence, Designing in a pandemic to fight a pandemic. “I was thrilled when Andy asked me to join him on this project,” says Sutherland. Computers tried but had been unable to crack the problem. (i.e., the 42nd year of the ninth century), was the last. The author’s choice of the number 42 has become a fixture of geek culture. They then used a number of optimizations and adaptations to make the code better suited for a massively distributed computation, compared to a computation run on a single supercomputer, says Sutherland. These numbers are extremely rare, much more so than prime numbers: only 14 of the former are lower than one billion. The puzzle is an illustration consisting of 42 multi-coloured balls, in 7 columns and 6 rows. Here you will read about a popular alternate theory and where and how Douglas Adams explained that it is very smart but not what he had had in mind. Booker and Sutherland discussed the algorithmic strategy to be used in the search for a solution to 42. In other words, the cube of an integer modulo 9 is –1 (= 8), 0 or 1. The work is far from over. The paper reporting their findings concluded that the question of the existence of a solution for numbers below 1,000 remained open only for 33, 42, 74, 114, 165, 390, 579, 627, 633, 732, 795, 906, 921 and 975. The Lowest Crime Rates in 42 Douglas Adams. In two years, the MIT Quest for Intelligence has allowed hundreds of students to explore AI in its many applications. The first practical numbers are 1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 48, 54, 56, 60, 64, 66 and 72 (sequence A005153 in OEIS). Here is how a perfectly ordinary number captured the interest of sci-fi enthusiasts, geeks and mathematicians, Everyone loves unsolved mysteries. The sequence corresponds to entry A020988 in The On-Line Encyclopedia of Integer Sequences (OEIS), created by mathematician Neil Sloane. If solutions can be found for all those examined values, it will be reasonable to conjecture that for any integer n that is not of the form n = 9m + 4 or n = 9m + 5, there are solutions to the equation n = a3 + b3 + c3. This sum of three cubes puzzle, first set in 1954 at the University of Cambridge and known as the Diophantine Equation x3+y3+z3=k, challenged mathematicians to find solutions for numbers 1-100. You cannot get a sum of 4 or 5 (= –4). Ryan Mandelbaum writes for Gizmodo about the efforts of principal research scientist Andrew Sutherland and others to use a crowd-sourced supercomputer to solve a math problem that scientists have been working on since the 1950s. In one instance, the series suggests that 42 is the answer to the question “What do you get if you multiply six by nine?” That idea seems absurd because 6 x 9 = 54. As a result, the carbon footprint of this computation — related to the electricity our computations caused the PCs in the network to use above and beyond what they would have used, in any case — is lower than it would have been if we had used a supercomputer.”. Read on to find out more about the whole situation. In 1936, for example, Alan Turing showed that no algorithm can solve the halting problem for every possible computer program. Such is the enduring interest in Douglas Adams’s story that it is due to be adapted into a new television series by Hulu, a streaming service. Douglas Adams, the “master of wacky words and even wackier tales” (Entertainment Weekly) once again boggles the mind with a completely unbelievable story of ghosts, time travel, eccentric computer geniuses, Samuel Taylor Coleridge, the end of the world, and—of course—missing cats. But when he turned to solve for 42, Booker found that the computing needed was an order of magnitude higher and might be beyond his supercomputer’s capability. In base 2, the nth element may be specified by repeating 10 n times (1010 ... 10). Since the first such school was created in France in 2013 there has been a proliferation of private computer-training institutions in the “42 Network,” whose name is a clear allusion to Adams’s novels. There have been many theories from fans in an attempt to explain why the number 42 was chosen, and what secret meaning it may hold. As if.) When it was applied to the “sum of three cubes” problem, 42 was more troublesome than all the other numbers below 100. “42” Is the meaning of life, the universe and everything. in English litera… For any integer p: By multiplying each term of these equations by the cube of an integer (r3), we deduce that there are also infinitely many solutions for both the cube and double the cube of any integer. Structural biologist Pamela Björkman shared insights into pandemic viruses as part of the Department of Biology’s IAP seminar series. Adding any three numbers among these numbers gives: 0 = 0 + 0 + 0 = 0 + 1 + (–1); 1 = 1 + 0 + 0 = 1 + 1 + (–1); 2 = 1 + 1 + 0; 3 = 1 + 1 + 1; 6 = –3 = (–1) + (–1) + (–1); 7 = –2 = (–1) + (–1) + 0; 8 = –1 = (–1) + 0 + 0 = 1 + (–1) + (–1). All were eventually solved, or proved unsolvable, using various techniques and supercomputers, except for two numbers: 33 and 42. Catalan numbers were first mentioned, under another name, by Swiss mathematician Leonhard Euler, who wanted to know how many different ways an n-sided convex polygon could be cut into triangles by connecting vertices with line segments. “When you're sitting at your computer reading an email or working on a spreadsheet, you are using only a tiny fraction of the CPU resource available, and the Charity Engine application, which is based on the Berkeley Open Infrastructure for Network Computing (BOINC), takes advantage of this. “There are four very easy solutions that were known to the mathematician Louis J. Mordell, who famously wrote in 1953, ‘I do not know anything about the integer solutions of x3 + y3 + z3 = 3 beyond the existence of the four triples (1, 1, 1), (4, 4, -5), (4, -5, 4), (-5, 4, 4); and it must be very difficult indeed to find out anything about any other solutions.’ This quote motivated a lot of the interest in the sum of three cubes problem, and the case k=3 in particular. “It’s very gratifying.”. ", International team launches vast atlas of mathematical objects, More about MIT News at Massachusetts Institute of Technology, Abdul Latif Jameel Poverty Action Lab (J-PAL), Picower Institute for Learning and Memory, School of Humanities, Arts, and Social Sciences, View all news coverage of MIT in the media. The formula for this sequence is a(n) = (2/3)(4n – 1). Take author Douglas Adams’s popular 1979 science-fiction novel The Hitchhiker’s Guide to the Galaxy, the first in a series of five. The number 42 also appears in different forms in the film Spider-Man: Into the Spider-Verse. To illustrate how difficult it is to find solutions to the equation n = a3 + b3 + c3, let’s see what happens for n = 1 and n = 2. Apart from allusions to 42 deliberately introduced by computer scientists for fun and the inevitable encounters with it that crop up when you poke around a bit in history or the world, you might still wonder whether there is anything special about the number from a strictly mathematical point of view. Sutherland is also a fan of Douglas Adams, so the project was irresistible. “One then enumerates values for d and checks each of the possible x, y, z associated to d. In the attempt to crack 33, the search bound B was 1016, but this B turned out to be too small to crack 42; we instead used B = 1017 (1017 is 100 million billion). That is, are there three cubes whose sum is 42? A team led by Andrew Sutherland of MIT and Andrew Booker of Bristol University has solved the final piece of a famous 65-year old math puzzle with an answer for the most elusive number of all: 42. “This is another reason I really liked running this computation on Charity Engine — we actually did use a planetary-scale computer to settle a longstanding open question whose answer is 42.”. Subscribers get more award-winning coverage of advances in science & technology. Douglas Adams FIN42 is a star system in No Man's Sky. The number 42 is especially significant to fans of science fiction novelist Douglas Adams’ “The Hitchhiker’s Guide to the Galaxy,” because that number is the answer given by a supercomputer to “the Ultimate Question of Life, the Universe, and Everything.”. In a 2016 preprint paper, Sander Huisman, now at the University of Twente in the Netherlands, pressed on and found a solution for 74: (–284,650,292,555,885)3 + (66,229,832,190,556)3 + (283,450,105,697,727)3. This triplet space is therefore infinite, unlike the computation for the sum of squares. The reference to base 13 in Adams’s answer requires a more indirect explanation. The characters tasked with getting that answer are disappointed because it is not very useful. In the binary system, or base 2, 42 is written as 101010, which is pretty simple and, incidentally, prompted a few fans to hold parties on October 10, 2010 (10/10/10). All orders are custom made and most ship worldwide within 24 hours. Like other computational number theorists who work in arithmetic geometry, he was aware of the “sum of three cubes” problem. 2. Hitchhiker’s conspiracy theorists cite Lewis Carroll as an influence in Adams using 42 as the answer to life, the universe and everything. Jean-Paul Delahaye is a professor emeritus of computer science at the University of Lille in France and a researcher at the Research Center in Computer Science, Signal and Automatics of Lille (CRIStAL). 42 is a special number because it is the funniest of all numbers. In 2009, employing a method proposed by Noam Elkies of Harvard University in[[OR: by American mathematician Noam Elkies in 2000, German mathematicians Andreas-Stephan Elsenhans and Jörg Jahnel explored all the triplets a, b, c of integers with an absolute value less than 1014 to find solutions for n between 1 and 1,000. The number 42 also turns up in a whole string of curious coincidences whose significance is probably not worth the effort to figure out. Demonstrating this assertion is straightforward: we use the “modulo 9” (mod 9) calculation, which is equivalent to assuming that 9 = 0 and then manipulating only numbers between 0 and 8 or between −4 and 4. An obvious question, which indeed has been asked, is whether the use of 42 in Adams’s books had any particular meaning for the author. Douglas himself attended Brentwood School after passing their entrance exam before going onto learn at the prestigious Cambridge University where he graduated with a B.A. © 2021 Scientific American, a Division of Springer Nature America, Inc. Support our award-winning coverage of advances in science & technology. Douglas Adams has said, For p = 1, we get: Note that for n = 3, as of August 2019, only two solutions were known: A question that naturally follows is: Is there at least one solution for every nonprohibited value? When the email from Charity Engine arrived, it provided the first solution to x3+y3+z3=42: 42 = (-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3, “When I heard the news, it was definitely a fist-pump moment,” says Sutherland. That calculation is not the only other solution. The difficulty appears so daunting that the question “Is n a sum of three cubes?” may be undecidable. The number 42 was difficult, but it is not the final step! As a region, 42 Douglas Adams is ranked 3,654 th in the world for Lowest Crime Rates. The Gutenberg Bible, the first book printed in Europe, has 42 lines of text per column and is also called the “Forty-Two-Line Bible.”, According to a March 6 Economist blog post marking the 42nd anniversary of the radio program The Hitchhiker’s Guide to the Galaxy, which preceded the novel, “the 42nd anniversary of anything is rarely observed.”, An obvious question, which indeed has been asked, is whether the use of 42 in Adams’s books had any particular meaning for the author. Still, a recent news item caught their attention. Principal research scientist Andrew Sutherland and a colleague at the University of Bristol have solved a decades-old math problem known as the “summing of the three cubes.” The team found the answer with the help of the platform the Charity Engine, “which utilizes idle, unused computing power from over 500,000 home PCs to create a crowdsourced and environmentally conscious supercomputer,” writes David Grossman for Popular Mechanics. Why couldn't Bristol's supercomputer solve this problem? If, for example, you ask your search engine variations of the question “What is the answer to everything?” it will most likely answer “42.” Try it in French or German. Explore our digital archive back to 1845, including articles by more than 150 Nobel Prize winners. In 2019 Andrew Booker of the University of Bristol in England settled the case of 33: 8,866,128,975,287,528)3 + (–8,778,405,442,862,239)3 + (–2,736,111,468,807,040)3. “Well, any computer *can* solve the problem, provided you are willing to wait long enough, but with roughly half a million PCs working on the problem in parallel (each with multiple cores), we were able to complete the computation much more quickly than we could have using the Bristol machine (or any of the machines here at MIT),” says Sutherland. If we could prove such undecidability, that would be a novelty. As a practical matter, the difficulty in making this calculation is that for a given n, the space of the triplets to be considered involves negative integers. Booker says he received many offers of help to find the answer, but instead he turned to his friend Andrew "Drew" Sutherland, a principal research scientist in the Department of Mathematics. For integers below 1,000, only 114, 390, 579, 627, 633, 732, 921 and 975 remain to be solved. Some propose that it was chosen because 42 is 101010 in binary code, others have pointed out that light refracts through a water surface by 42 degrees to create a rainbow, and others have commented that light requires 10−42 seconds to cross the diameter of a proton. Associate Professor Michael Short's innovative approach can be seen in the two nuclear science and engineering courses he’s transformed. From that point, Douglas Adams’s number was the last positive integer lower than 100 whose representation as a sum of three integer cubes was unknown. We thus say that n = 9m + 4 and n = 9m + 5 are prohibited values. Often you hear it as a simple namecheck, a gently conclusive "Douglas Adams", … Such is the case for all integers n that are expressible as 9m + 4 or 9m + 5 for any integer m (e.g., 4, 5, 13, 14, 22, 23). Born in 1952 on the 11th of March, Douglas Adams was brought up in East London after his family moved there from Cambridge. “All opinions are not equal. For the sum of cubes, some solutions may be surprisingly large, such as the one for 156, which was discovered in 2007: 156 = 26,577,110,807,5693 + (−18,161,093,358,005)3 + (−23,381,515,025,762)3. It seems that Douglas Adams was right after all: the answer to Life, the Universe and everything, is 42. 24. Moreover our interest holds even if the mystery is based on a joke. And the two had worked together before, helping to build the L-functions and Modular Forms Database (LMFDB), an online atlas of mathematical objects related to what is known as the Langlands Program. Discover world-changing science. Otherwise, the main difference between the search for 33 and the search for 42 would be the size of the search and the computer platform used. “There is a single integer parameter, d, that determines a relatively small set of possibilities for x, y, and z such that the absolute value of z is below a chosen search bound B,” says Sutherland. But here we are in a readily describable, purely mathematical domain. The research carried out thus far, which depends on the power of the computers or computer networks used, has produced an ever expanding body of results. Scientific American is part of Springer Nature, which owns or has commercial relations with thousands of scientific publications (many of them can be found at, the 42nd anniversary of anything is rarely observed, The On-Line Encyclopedia of Integer Sequences, explore the properties of various numbers. And like the two preceding sequences, the density of numbers is null at infinity. 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