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Math

60-Year-Old Maths Problem Partly Solved By Amateur (theguardian.com) 161

An amateur mathematician has made the first breakthrough in more than 60 years towards solving a well-known maths problem. From a report: Aubrey de Grey, who is more widely known as a maverick biologist intent on extending the human lifespan, has taken the academic world by surprise after announcing a new solution to the so-called Hadwiger-Nelson problem. The problem sounds deceptively simple, but despite some professionals spending years trying to crack it, progress has stalled since shortly after the puzzle was first posed in 1950. "Literally, this is the first progress in more than 60 years," said Gil Kalai, a mathematician at Hebrew University of Jerusalem.

The problem is as follows. Imagine a collection of dots connected by lines. The dots can be arranged any way at all, the only rule is that all the connecting lines must be of equal length. For instance, in a square the diagonal would not be joined up, but the outer edges would be. Now, colour in all the dots so that no two connected points have the same colour. How many colours are required. For a square, the answer would be two. But the Hadwiger-Nelson problem asks what the minimum would be for any configuration -- even one that extends across a plane of infinite size.

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60-Year-Old Maths Problem Partly Solved By Amateur

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  • by zmaragdus ( 1686342 ) on Friday May 04, 2018 @11:27AM (#56554028)
    • by msmash ( 4491995 ) Works for Slashdot
      Thanks. Due to some unicode issue, the correct Wikipedia link is breaking in the summary. An alternative would be using a URL shortner, but I think many people would not want to have a secondary link, so I have swapped the link with a different source altogether.
      • Wikipedia uses, in its infinite wisdom, the character U+2013 (en-dash) in article names like those. You get the correct link to the article with decoded characters when you copy the link from the tab heading "Article" at the top of the article. It shows the en-dash as %E2%80%93, as it should.
  • Jesus (Score:1, Insightful)

    by Anonymous Coward

    I almost fell asleep reading that summary. Is this of any practical use what so ever? What is that â" in the last sentence supposed to be?

    • Re:Jesus (Score:5, Insightful)

      by jellomizer ( 103300 ) on Friday May 04, 2018 @11:44AM (#56554156)

      It could, the geometric shape can be abstracted into other things. Such as time, and process, I could see this being used to help optimize a parallel programming process where a particular code takes so long to run, but to avoid collisions the different elements in time, will need to hit different process points at different time, then you need to span it up, so how many unique process points will you need. Say with a massively parallel system, such as an advanced quantum computer.

      • by mikael ( 484 )

        That would be Hilbert space filling curves. Those are used to optimize parallel processing. Also used in a way to arrange the folding of the brain so that every region is a minimum distance from the spinal cord and optic nerves.

        • by sfcat ( 872532 )

          That would be Hilbert space filling curves. Those are used to optimize parallel processing. Also used in a way to arrange the folding of the brain so that every region is a minimum distance from the spinal cord and optic nerves.

          Yes, but this is a specialization of the problems that filling curves handle. Specifically when the edges need to be straight and the same length. Filling curves handles the general case but potentially this would be a better solution for a unique case with this extra constraint. Seems possible that some manufacturing and design problems have this constraint as then the "edges" (wires or some such) could all be the same. When filling curves is used, its likely those "edges" would need to be different an

      • Processing nodes, but also learning nodes, fractal analysis, curve-fitting models, and of course, trickle-down economics :-)

    • Re:Jesus (Score:5, Insightful)

      by gtall ( 79522 ) on Friday May 04, 2018 @12:30PM (#56554504)

      Hmmm...lemme guess, when Evariste Galois was inventing group theory, you'd be in the peanut gallery claiming you couldn't see any use for it (hint, he did it in the early 1800s). And when those wild and crazy guys were screwing up developing the math for quantum mechanics, you'd be asking them for a practical use (hint, they did it in the early 1900s).

      In short, if someone cannot point to a use of something, it shouldn't be done. How enlightened of you? Have you explained your theory to scientists? I'm sure they'd listen to you.

    • Heck, I once partly solved a great maths problem too, the conjecture that x^n + y^n = z^n has no non-zero integer solutions for x, y and z when n > 2. My partial solution was "let x be a nonzero positive integer".
  • Not quite (Score:3, Informative)

    by Anonymous Coward on Friday May 04, 2018 @11:32AM (#56554052)

    If you read the articles, he pushed the instance of contradictory evidence from N=4 to N=5, but has no proof that N=6 isn't a instance.

    Thus, it is a new piece of evidence that N>=5, but not that it is solved.

  • by Kenja ( 541830 ) on Friday May 04, 2018 @11:34AM (#56554084)
    it does not mean you're any good at what you do.
  • by tgibson ( 131396 ) on Friday May 04, 2018 @11:35AM (#56554100) Homepage

    Aubrey De gray's finding has the attention of the Polymath Project [wikipedia.org], "a collaboration among mathematicians to solve important and difficult mathematical problems by coordinating many mathematicians to communicate with each other on finding the best route to the solution."

    You can follow their current conversation here [wordpress.com].

    • Sorry, I mean "Aubrey de Grey"

    • by deesine ( 722173 )

      Interesting, according to Aubrey "I got lucky and [Dan P. Ismailescu] got unlucky". Also, the "problem was ripe for the picking. And while we like your proof better, we are very happy that ours is different." says Dan P. Ismailescu about teaming with Geoffrey Exoo for three years. Refreshing the respect these guys show each other.

      Think I'm done reading /. today, thanks for the link.

  • by will_die ( 586523 ) on Friday May 04, 2018 @11:54AM (#56554204) Homepage
    https://www.quantamagazine.org... [quantamagazine.org]

    Is the article article about what was done, not the cut down version from a gossip rag sheet which is given in the summary.
  • Big Science and maverick outsiders can actually greatly benefit from each others' existences, for the mavericks are free to ask questions which might be out-of-bounds in academic circles -- and so, they are much better positioned to start new groundbreaking lines of investigation. We can solve the publish-or-perish problem with this exact approach, but it will require us to care more about long-term innovations than our short-sighted desires to confirm our pre-existing worldviews.

    The top-down philosophical

    • by gtall ( 79522 )

      "they are much better positioned to start new groundbreaking lines of investigation" I do not think so. Science these days requires lots of years of experience. You get the one or two odd contributions from outsiders, but largely scientific progress is made in the trenches slogging it out year after year. Care to explain any advances in string theory due to outsiders to physics?

    • by lgw ( 121541 )

      Well, I don't know how generally true that is, but certainly in Math there is a class of old, unsolved problems that professors don't want to be seen as working on, as they are assumed to be a waste of time and thus it's somewhat embarrassing to be working on them. Obviously, that concern doesn't apply to anyone who doesn't have "math professor" as their day job.

  • Perhaps this could lead to a proof for the somewhat similar four color theorem? Also deceptively simple, and maybe obvious, but unproven.
    • by nasch ( 598556 )

      Perhaps this could lead to a proof for the somewhat similar four color theorem?

      No need.

      "The four color theorem was proved in 1976 by Kenneth Appel and Wolfgang Haken... To dispel remaining doubt about the Appel–Haken proof, a simpler proof using the same ideas and still relying on computers was published in 1997 by Robertson, Sanders, Seymour, and Thomas. Additionally, in 2005, the theorem was proved by Georges Gonthier with general-purpose theorem-proving software."

      https://en.wikipedia.org/wiki/... [wikipedia.org]

  • So basically the problem seems to come to what is the most "connection dense setup". How many connections can you give every dot. But then that is not necessarily the entire solution, as odd vs even can have an effect as a triangle with an odd number of dots in a circular connection needs more colors than a square. I can see why they limit this to 2d planes, as the answer will clearly go up with every dimension added. 1D is simple, the answer is always 2 (for dots>1). Honestly, it sounds like it should b

    • In 3D the number most likely jumps to infinity. This is like the how many colours does it take to colour a map so that no adjacent countries have the same colour. 1D is trivally 2, 2D is four but the proof sucks, 3D is clearly infinity.
      • I don't know. In this case 3D is infinite because countries can be any shape (so you can have an infinite number of countries next any one country). In the problem suggested, all lines are of equal length, so the countries all have to be the same size. This to me seems like it would make the solution be a fairly reasonable number. In 2d their is clearly going to be a fairly low number of max connections to a single dot if you also want those dots connected in a circle pattern. And while 3D turns that circle

        • Imagine the 2D solution is 5. I then likely have a graph with 5 colours where two points not adjacent must be the same colour. Take this graph, copy it and then rotate it slightly out of the plane about one of those two points until the other point is the line distance away from it. Now those points are connected and then must be of opposite colors so I have a graph that now requires 6 colors. Also this new graph likely has 2 points that must be the same color. I copy and rotate again creating somethin
          • Before someone asks, no, the rotation about a point doesn't work in 2D since in rotating about the first point your second point will end up in the same place as an existing point and so will many of the other points in the graph. Where as I can always rotate the required distance in 3D and not end up on an existing point.
          • Exactly, but don't we have a countable number of rotations, therefore necessitating a countable solution?

            But I am starting to see why this problem is so hard. If i understand it correctly, dots can be any distance apart, it is just that the ones equal distance apart have lines.

            So the solution is probably not a pretty looking pattern, where all neighbor dots are equal distance but instead a bloody mess of thousands of patterns overlapping each other and all you can see is chaos.

      • by slew ( 2918 )

        In 3D the number most likely jumps to infinity. This is like the how many colours does it take to colour a map so that no adjacent countries have the same colour. 1D is trivally 2, 2D is four but the proof sucks, 3D is clearly infinity.

        Although it might be tempting to "analogize" the problem the 4 color map problem, in fact the problems are not at all similar and have a different answer.

        Even, the wikipedia entry [wikipedia.org] on this problem has an answer to this particular generalization to 3D...

        The problem can easily be extended to higher dimensions. In particular, finding the chromatic number of space usually refers to the 3-dimensional version. As with the version on the plane, the answer is not known, but has been shown to be at least 6 and at most 15.

        This is a pointer [cs.bme.hu] to paper the illustrates the upper bound for R^3 in case you are interested.

  • hmmm (Score:2, Informative)

    by nomadic ( 141991 )

    So looking at his Wikipedia page, he doesn't sound like an amateur mathematician if he has a degree in computer science. Interestingly, he seems like an amateur biologist in the sense that he was self-taught and awarded a PhD based on a book he wrote based on that self-teaching.

    • by lgw ( 121541 )

      "Professional" means "you get paid to do this", and nothing else. He's not an math prof, so he's an amateur.

      • by nomadic ( 141991 )

        If he gets paid to program, that's a subfield of math.

        • by lgw ( 121541 )

          Only in the same way that math is a subfield of biology, since it's done by humans.

          He's not paid to publish math papers, is the point.

      • "Professional" means "you get paid to do this", and nothing else. He's not an math prof, so he's an amateur.

        No, "professional" means you're a member of a government-regulated "profession". A lawyer who isn't practicing is still a professional.
        As is an electrician, a doctor, an engineer (the one that drives a train engine), a contractor, a pilot, etc.

        Things that aren't professionals include professors, software "engineers", most other engineers (they're weakly and not universally regulated), and athletes.

        • "Professional" means "you get paid to do this", and nothing else. He's not an math prof, so he's an amateur.

          No, "professional" means you're a member of a government-regulated "profession". A lawyer who isn't practicing is still a professional. As is an electrician, a doctor, an engineer (the one that drives a train engine), a contractor, a pilot, etc.

          Things that aren't professionals include professors, software "engineers", most other engineers (they're weakly and not universally regulated), and athletes.

          Care to provide a citation for any of that? The closest definition here [dictionary.com] has "learned profession" as originally referring to theology, law, and medicine.

          • It's in the word itself. Professing is all about declaring something publicly and that being recognized officially.

            It comes from profiteri which is to declare, testify, attest, etc. publicly and openly. Later, the word "profess" got tied up taking vows in the Church, and then similarly adopted for people taking oaths to become members of various trade guilds.

            Ultimately, you can't be a professional unless what you do is openly attested to and officially recognized. In today's world, that's being regulated

            • by lgw ( 121541 )

              No matter how much sense a definition makes in your own head, that's not how language works.

              • Except that's literally the definition, but ok.

                • by lgw ( 121541 )

                  Looks like the kind of problem that can only be settled by a statistical survey of dictionaries! Your definition seems to be about half of them, so half the population are wrong-headed freaks such as yourself. Amusingly, this one [yourdictionary.com] explicitly includes higher education.

            • It's in the word itself. Professing is all about declaring something publicly

              Yes, I saw "occupation one professes to be skilled in" as part of the origin of the word.

              and that being recognized officially.

              I don't see that anywhere.

              • Try looking up the word and its use throughout history?
                Try looking up laws in your area regulating various professions?
                See which professions are regulated and which aren't? Almost universally in the western world, medicine and law are regulated professions. Being a "software engineer" or a professor at a school are not.

                • Try looking up the word and its use throughout history?

                  I looked it up in a dictionary. That's where most people go to find the definition of a word.

                  Try looking up laws in your area regulating various professions? See which professions are regulated and which aren't? Almost universally in the western world, medicine and law are regulated professions. Being a "software engineer" or a professor at a school are not.

                  "Some professions are regulated" does not imply "Professions must be regulated".

            • Wow, I'm surprised to see somebody in the wild who mistakes a words etymology for its definition.

              Even sadder is conflating "declaring publicly" with "recognized officially." As if History has never taught the difference! LMFAO You can find extensive examples just from the Greeks, you don't even have to resort to the Current Era.

              • It's not a mistake, it's an understanding of the word, its origin, and its use.
                I didn't conflate anything, go look it up. The word isn't just about saying something, it's about attesting to it.

        • by ceoyoyo ( 59147 )

          I was required to take a professional ethics course as part of my undergraduate degree. Your definition is indeed one of the accepted definitions. There are others, including the OP's. Another is that you belong to some recognizable group that adhere's to some acknowledged code. When I got my BSc and PhD I swore to be bound by the rights and responsibilities of the degree etc., which could qualify.

          The course textbook had about a hundred pages on the subject. Which was far more interesting then the actu

          • But where do you come from where the engineers (not the train driving ones) are weakly regulated!? Is it in Florida where that bridge collapsed?

            I'm not aware of any regulatory bodies overseeing the "software engineers" and their work, for example.

            Similarly for other engineers. The only group of engineers I can think of that is regulated to any actual efficacy are civil engineers, due to their involvement in public roads and structures. Mechanical engineers, electrical engineers, etc. usually don't have any state-wide regulation outside of what applies to their trade in general, as far as I know. An "electrical engineer" in CA is still going to b

            • by ceoyoyo ( 59147 )

              I think the US must be some kind of special case. Engineers are absolutely regulated here. A professional engineer must pass tests, work under another PEng for a period of time, pay annual dues to a professional association, has particular legal liabilities, and the title itself is regulated. Electricians are not electrical engineers, and programmers are not software engineers.

              • by lgw ( 121541 )

                The US has "Professional Engineer" as a separate thing. It's kind of a joke to qualify for, but the point of it is you lose it if you sign of on something stupid, so it does serve a useful function. BTW, "electrician" and "electrical engineer" are unrelated fields in the US. Electricians memorize the hundreds of pages of the national electric code [nfpa.org], while EEs design circuit boards. Any idiot with a degree can be an EE, but a master electrician is a professional. Sounds like the rest of the world has it

        • Interesting definition of "professional".
          I go with your parent ...
          But I'm from Europe :D

    • by hey! ( 33014 )

      A BSCS only gets you to the foothills of that particular mountain range.

  • by JoshuaZ ( 1134087 ) on Friday May 04, 2018 @12:02PM (#56554284) Homepage
    It was well known that one never needs more than 7 colors- this comes from a little work where one can tile the plane with hexagons and then pick 7 distinct colors cleverly. It was also known that you needed at least 4 colors, since one can construct configurations which require 4 colors. Both of these parts are simple enough that working out the details are fun exercises. What Aubrey de Grey did is use a careful construction involving certain specific subconfigurations to aid a computer search to construct a very big configuration which was highly likely to need 5 colors; he then verified using a computer system that this configuration did require 5 colors. But note that while this is progress this isn't a full solution; this shows that the number of colors needed is at least 5, but whether it is 5,6 or 7 is unclear. My own guess based on his work is tentatively towards 6 because it looks like there's a lot of room in his configuration that might allow one to bump it up to 6 with a few more ideas and the argument for why one needs at most 7 is so simple that it seems like something should be able to reduce that even if no one has figured it out yet.
  • by sjbe ( 173966 ) on Friday May 04, 2018 @12:03PM (#56554302)

    An amateur mathematician has made the first breakthrough in more than 60 years towards solving a well-known maths problem.

    Why is it relevant whether he gets paid to solve mathematical problems or not? Amateur just means that someone doesn't derive any income from the task. It has nothing to do with competence or the lack thereof. Plenty of people are very talented at things they don't get paid for.

    • I guess Einstein was a 'amateur physicist" when he came up with theory of relativity
      • by tomhath ( 637240 )
        Nope. He was hired by the patent office as an expert to testify in court on things like inertial navigation. The legend that he was a humble clerk is just that, a legend; at that point in his life he had a PhD in physics was was well known internationally.
    • It's relevant because someone paid to do mathematics is presumed to have the time, inclination, motivation, and ability to advance the field. An amateur is presumed to only be able to work on problems in his scant spare time, with a mind trained to handle problems in another field. This implies either that the amateur is a truly superior intellect or that the professionals are slackers.

      Not correct, of course, but it's similar to rooting for the underdog.

      • It's relevant because someone paid to do mathematics is presumed to have the time, inclination, motivation, and ability to advance the field.

        That would be a naive assumption. It's not at all unusual to find scientists and engineers who are more than fluent in some rather arcane branches of mathematics. Math is the language of science and engineering. Almost any professional physicist is going to be highly competent at mathematics. Should it really be surprising that some of them might spend a bit of time working on some random math puzzles in their spare time or that they might be pretty good at it? Where they derive their income should be a

    • I'd say that the difference here is that a professional mathematician will exist in a certain environment. A math professor will read certain journals, associate with other math professors, teach certain things to students, etc. An amateur will be outside this environment.

      Currently, it's really difficult to get up to speed in a science without being a professional (this wasn't the case if you go back far enough), and it's really difficult to make contributions without knowing the existing science very

      • by epine ( 68316 )

        I'd say that the difference here is that a professional mathematician will exist in a certain environment. A math professor will read certain journals, associate with other math professors, teach certain things to students, etc. An amateur will be outside this environment.

        Absolutely the crucial connotation.

        Amateur, especially in egghead ventures, almost always implies a lone wolf, or at most the Chudnovsky brothers' hillbilly pen pals (five precocious amateurs, each one that much weirder than the last).

    • People should care about amateur status. It deserves to be elevated above the same achievement of a professional. When amateurs achieve something professionals do not it becomes evidence that achievements in a field are borne out of talent rather than grinding. It shows that you can achieve without funding and fancy equipment.

      • People should care about amateur status. It deserves to be elevated above the same achievement of a professional.

        I don't agree. First off all, "amateurism" is something of an overblown myth. Just because you don't get paid to do something doesn't mean you haven't put in a huge amount of time and effort. A lot of Olympic athletes are "amateurs" because they don't get paid to play but make no mistake that they've devoted a good portion of their life to their chosen sport and are very very good at it. Second, the achievement deserves to stand on its own merits. Why should someone who devoted their life to a vocation

        • First off all, "amateurism" is something of an overblown myth. Just because you don't get paid to do something doesn't mean you haven't put in a huge amount of time and effort.

          Indeed, but there's few if any people who were able to put in 40hours a week into their hobby.

          A lot of Olympic athletes are "amateurs" because they don't get paid

          That hasn't been true on any kind of reasonable scale since the last world war, and those "Olympic" athletes that remain mostly are there not there based on skill, hell there's a share of them that are lucky not to drown in the swimming pool (minimum number of entrants for countries, points systems that get gamed by selecting your fights).

          Why should someone who devoted their life to a vocation and happens to get paid for it be more or less worthy of accolades than someone who derives their income from some other profession? That makes zero sense.

          Resources matter. Simply claiming they don't doesn't make it so.

          There is nothing about talent that is more worthy of respect than there is about hard work.

          I didn't say

  • by Anonymous Coward

    Essentially, it has been known for a while that the answer is either 4 or 5 or 6 or 7 [wolfram.com].

    This paper identifies a graph that cannot be colored with just 4 colors, so it establishes 5 as the new lower bound.

    • by jrumney ( 197329 )
      Maybe I'm missing a rule here, but I can trivially put down some dots making a bunch of diamond shapes radiating from a central point such that a lot more than 7 colours are required to give unique colour to all the dots connected to the centre dot. It says the diagonals are not connected for a square, so I don't see a requirement that they be connected for a diamond either, and such a rule seems to be the only thing making a plane of hexagons the bounding shape.
      • by jrumney ( 197329 )
        Having looked at the Wikipedia page now, I see that I misinterpreted the problem. If I have say 10 lines radiating out from a common centre, it doesn't matter if the other ends are all the same color, as they cannot join to each other. Each line is considered independently, not together with all the lines connecting to the same dot.
  • While it is nice, that you seem to have time for a hobby, the rest of us would prefer that you concentrate your mental power to solve the problem of immortality.
    Thanks in advance

    • If you had a decent education, you would know, your soul is immortal, and only your flesh will perish (and jokes aside: that is the mantra of most (if not all?) religions).

Logic is the chastity belt of the mind!

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