7 ms·

Great read! When you first hear the taxicab number story, your initial impression is to be struck by Ramanujan's innate calculating capability. It's interesting

by v64 7y ago

Great read! When you first hear the taxicab number story, your initial impression is to be struck by Ramanujan's innate calculating capability. It's interesting to find out that the real coincidence here is that Hardy rode in a taxicab whose number had happened to show up in Ramanujan's investigations of Fermat's last theorem.

hnews_account_1 7y ago
A lot of genius stories are like this. I was also under the illusion that these guys could just do things that fast, but at some point, I read Feynman's biography where he explicitly talks about how he used to solve homework problems or something beforehand and then he used to pretend that he found the solution while solving it if his classmates asked. That threw me for a loop and I started believing shit like no one's smarter than I was etc. Then I just ... grew up, I guess. And I remembered this story by Feynman and I realised that despite his absolutely undoubtable genius, he'd have appeared godlike to me if I was his classmate back in the day. Ramanujan's brain worked even faster by most accounts. He dreamed in math, I think. So there are multiple stories where people ask him a puzzle and he'll answer with an equation that solves it for the entire family of problems that the puzzle could come from.
- dorchadas 7y ago
  What was the quote about Feynman? That he loved to cultivate anecdotes about himself or something similar? Makes a lot of his stories make a lot more sense, too.
  - avip 7y ago
    It's a rant by fellow Physicist Gell-Mann.
  - Quekid5 7y ago
    I think it's pretty plausible that he was a raging egomaniac (narcissist, perhaps?). Undoubtedly a great thinker and genius, but that doesn't say very much about personality traits.
    - deleted 7y ago
      [deleted]
  - user2994cb 7y ago
    Feynman even has his own 1729 anecdote: https://www.ee.ryerson.ca/~elf/abacus/feynman.html https://www.ee.ryerson.ca/~elf/abacus/feynman.html
  - OldGuyInTheClub 7y ago
    From Murray Gell-Mann: https://www.youtube.com/watch?v=rnMsgxIIQEE https://www.youtube.com/watch?v=rnMsgxIIQEE Several clips from Gell-Mann's Web of Stories interview (late 1990s) pertain to his on-again off-again collaboration with Feynman. https://www.youtube.com/watch?v=o2sEW4ggVlA&list=PLVV0r6CmEsFxKFx-0lsQDs6oLP3SZ9BlA https://www.youtube.com/watch?v=o2sEW4ggVlA&list=PLVV0r6CmEs...
  - hnews_account_1 7y ago
    I think the part that makes it genuine is that he was comically self aware of himself and his craziness. Even when he was pushing the boundaries just for the sake of it and to make a caricature / character out of himself, he did it in a way that made me think that he didn't really pretend to not be doing it for his ego. It's like 4 levels of thinking somehow merged in his actions: 1) be normal and look at the crazy people, 2) be a crazy person, 3) be a crazy person and be aware of your craziness, 4) be a crazy person, be aware of it and let others know that you're aware of it. It feels like one of those thought spirals I go into if I have weed. It's right on the boundary of crazy but probably also (in his case) inside the realm of genius.
  - gameswithgo 7y ago
    i recall him explaining several shortcuts one can use to solve problems in seemingly impossible speeds by drawing on a breadth of experience from similar problems that you have memorized or are easy to compute and interpolating. its still genius but not in the sense of actually being able to do huge calculations in ones head the way a computer would.
    - deleted 7y ago
      [deleted]
    - hnews_account_1 7y ago
      Feynman has no dearth of stories showing his genius, but one specific example is a video 3Blue1Brown did of how Feynman converted the velocity vectors of a revolving body in a gravitational well (I think it was that) into a perfect circle of vectors. It's one of those results that's deep and yet you can find it yourself too if you spend enough time with it.
    - vidarh 7y ago
      Often it's also simply just that people are not used to thinking about more efficient ways of solving a problem. There's a (quite possibly apocryphal) story about Niels Henrik Abel in primary school, where his teacher supposedly wanted time to do some grading and assigned the students the busywork of adding up all numbers from 1 to 100. Abel supposedly quickly found the well known formula n(n+1)/2 and gave the teacher the answer within minutes, and the teacher supposedly believed he'd somehow "cheated" because he could not imagine any of them could figure it out. I have no idea if the story is real (I grew up in Norway, so Abel was a popular subject for stories like this) - it was told to me in high school by a maths teacher after giving us the modified task of seeing if we could find any shortcuts to doing the sums, and seeing what we'd come up with. I found the formula quickly, but at that age that's nothing special, especially not when prompted to find an alternative solution. But the overall idea the teacher was trying to get us to understand was how to pause and think about how to decompose a problem rather than just picking the most obvious alternative, and learning to be "lazy" in the sense of relentlessly looking for an easier way to do things is a large part of what got me into software development..
- karmakaze 7y ago
  This reminds me of the von Neumann fly puzzle story: https://en.wikipedia.org/wiki/John_von_Neumann#Cognitive_abilities https://en.wikipedia.org/wiki/John_von_Neumann#Cognitive_abi...
  - eesmith 7y ago
    Or swallow, if you think Wigner's account is more accurate than Halmos'.
  - hnews_account_1 7y ago
    Yeah, Neumann was well into the nutty end of the genius spectrum. As in, his genius was so nuts that it's hard to believe he was real. Similar with "The strangest man to ever visit my lab" - Paul Dirac (that's a quote from Neils Bohr). At least Dirac appeared to be autistic enough from the description that I'm not surprised he was able to do what he did. It's a superior intelligence, but I'm guessing he was obsessive about shit as well.
  - saagarjha 7y ago
    Of course, once you know how to sum infinite geometric series it's not too hard to come across the answer that way ;)
- wenc 7y ago
  I think most of us are impressed by computational parlor tricks (and indeed raw computational intelligence in general -- being able to process information and compute quickly and accurately), but for me, genius goes beyond that. Genius is about having rare and useful insights that the rest of us are incapable of, and that a computer is unable to easily replicate. For instance, there was this thing on Twitter recently about all percentages being reversible (7% of 50 is equal to 50% of 7, but the latter is easier to mentally calculate). Most of us are aware that multiplication is commutative, but it takes genius to recognize and frame that insight in a useful way.
  - cheerlessbog 7y ago
    It never occurred to me that 7% of 50 is equal to 50% of 7 perhaps because they are equally easy to calculate? Multiply 7 by 5 and fix the decimal point.
    - wenc 7y ago
      What about 17% of 50? 17*5 isn’t so simple anymore, whereas 50% of 17 is 8.5. To be fair, with most shortcuts, it’s possible to construct difficult cases (17% of 23 is difficult in either order) but where it applies (when one of the pairs is a common percentage), exploiting commutativity can be quite useful. Plus the mental overhead of remembering the rule is extremely minimal.
      - JadeNB 7y ago
        How can you discover that 50% of 17 is 8.5 if you can’t multiply 17*5? (If the answer is “by halving”, then my response is that halving and then multiplying by 10 is often the easiest way to multiply by 5!)
        Sharlin 7y ago
        I’m not sure, but to me at least halving 17 to get 8.5 is an almost completely intuitive process, whereas multiplying 17 by 5 seems to require an extra cognitive step to reduce it to shifting the decimal point and then halving (or vice versa). Never mind the extra step when presented by the problem of taking 17% of 50.
- Psyladine 7y ago
  Feynman was undoubtedly a genius, but he also suffered from a need to be admired. The safecracking episodes at Los Alamos are a perfect example - giving the impression he was an expert safe cracker when his real methodology was guesswork and sometimes subterfuge (birthdays, anniversaries, or even subtlety observing someone inputting their combination).
  - elSidCampeador 7y ago
    I get your point, I really do. But if what you call his "real methodology" worked well, why would he use more advanced/safecracker-y techniques? I'd say the real mistake he made was that he lifted the veil off of how he did things, leading people to say "oh even I could have done that".
    - Psyladine 7y ago
      Something something, invoice $1 for chalk, $9999 for knowing where to mark the 'X'.
      - elSidCampeador 7y ago
        zigackly (https://farm5.static.flickr.com/4096/4746820383_314f398f2f.jpg https://farm5.static.flickr.com/4096/4746820383_314f398f2f.j...)
- matt-noonan 7y ago
  David Kelly [1] once told me that as a grad student at Princeton, he somehow managed to get the Sunday New York Times delivered to him late Saturday night. He'd stay up all night solving the crossword puzzle, then dazzle everybody who was stumped on it the next day. [1] https://www.vinc17.net/yp17/index.en.html https://www.vinc17.net/yp17/index.en.html
zamadatix 7y ago
I'm not sure I'd call it coincidence rather "being struck by the vast amount of time and passion Ramanujan put into mathematics". I'd be absolutely amazed if he couldn't have recalled a similarly obscure fact for most numbers below 10,000.
macewindu 7y ago
> Ramanujan's innate calculating capability. Hard work and obsessive work effort on a specific area makes it appear innate.
mkl 7y ago
> When you first hear the taxicab number story, your initial impression is to be struck by Ramanujan's innate calculating capability. This is the way the story is always presented, and I think that's usually how it's intended, but I think it's quite misleading for another reason too. If you've ever made or looked at a table of cubes, the famous fact really jumps out (in base 10). I'm serious, look: n n³ -------- 1 1 2 8 3 27 4 64 5 125 6 216 7 343 8 512 9 729 10 1000 11 1331 12 1728 The two pairs of cubes are 1000 and 729, and 1728 and 1, and 1000 and 1 make the addition trivial and the similarity obvious (and 729 and 1000 are even right next to each other, one row away from 1728!). With that observation, it doesn't take much effort to try the smaller possibilities and see that 1729 is the smallest number that can be written as the sum of two cubes two different ways. Ramanujan knew numbers and their relationships intimately, better than Hardy, who knew more theory. I think Ramanujan knew the fact about 1729 already, and that you are right about the taxi number coincidence being more surprising and, well, impressive. (Yes, I've commented on this before: https://news.ycombinator.com/item?id=21165031 https://news.ycombinator.com/item?id=21165031)

by v64 7y ago

hnews_account_1 7y ago

A lot of genius stories are like this. I was also under the illusion that these guys could just do things that fast, but at some point, I read Feynman's biography where he explicitly talks about how he used to solve homework problems or something beforehand and then he used to pretend that he found the solution while solving it if his classmates asked. That threw me for a loop and I started believing shit like no one's smarter than I was etc. Then I just ... grew up, I guess. And I remembered this story by Feynman and I realised that despite his absolutely undoubtable genius, he'd have appeared godlike to me if I was his classmate back in the day. Ramanujan's brain worked even faster by most accounts. He dreamed in math, I think. So there are multiple stories where people ask him a puzzle and he'll answer with an equation that solves it for the entire family of problems that the puzzle could come from.

dorchadas 7y ago
What was the quote about Feynman? That he loved to cultivate anecdotes about himself or something similar? Makes a lot of his stories make a lot more sense, too.
- avip 7y ago
  It's a rant by fellow Physicist Gell-Mann.
- Quekid5 7y ago
  I think it's pretty plausible that he was a raging egomaniac (narcissist, perhaps?). Undoubtedly a great thinker and genius, but that doesn't say very much about personality traits.
  - deleted 7y ago
    [deleted]
- user2994cb 7y ago
  Feynman even has his own 1729 anecdote: https://www.ee.ryerson.ca/~elf/abacus/feynman.html https://www.ee.ryerson.ca/~elf/abacus/feynman.html
- OldGuyInTheClub 7y ago
  From Murray Gell-Mann: https://www.youtube.com/watch?v=rnMsgxIIQEE https://www.youtube.com/watch?v=rnMsgxIIQEE Several clips from Gell-Mann's Web of Stories interview (late 1990s) pertain to his on-again off-again collaboration with Feynman. https://www.youtube.com/watch?v=o2sEW4ggVlA&list=PLVV0r6CmEsFxKFx-0lsQDs6oLP3SZ9BlA https://www.youtube.com/watch?v=o2sEW4ggVlA&list=PLVV0r6CmEs...
- hnews_account_1 7y ago
  I think the part that makes it genuine is that he was comically self aware of himself and his craziness. Even when he was pushing the boundaries just for the sake of it and to make a caricature / character out of himself, he did it in a way that made me think that he didn't really pretend to not be doing it for his ego. It's like 4 levels of thinking somehow merged in his actions: 1) be normal and look at the crazy people, 2) be a crazy person, 3) be a crazy person and be aware of your craziness, 4) be a crazy person, be aware of it and let others know that you're aware of it. It feels like one of those thought spirals I go into if I have weed. It's right on the boundary of crazy but probably also (in his case) inside the realm of genius.
- gameswithgo 7y ago
  i recall him explaining several shortcuts one can use to solve problems in seemingly impossible speeds by drawing on a breadth of experience from similar problems that you have memorized or are easy to compute and interpolating. its still genius but not in the sense of actually being able to do huge calculations in ones head the way a computer would.
  - deleted 7y ago
    [deleted]
  - hnews_account_1 7y ago
    Feynman has no dearth of stories showing his genius, but one specific example is a video 3Blue1Brown did of how Feynman converted the velocity vectors of a revolving body in a gravitational well (I think it was that) into a perfect circle of vectors. It's one of those results that's deep and yet you can find it yourself too if you spend enough time with it.
  - vidarh 7y ago
    Often it's also simply just that people are not used to thinking about more efficient ways of solving a problem. There's a (quite possibly apocryphal) story about Niels Henrik Abel in primary school, where his teacher supposedly wanted time to do some grading and assigned the students the busywork of adding up all numbers from 1 to 100. Abel supposedly quickly found the well known formula n(n+1)/2 and gave the teacher the answer within minutes, and the teacher supposedly believed he'd somehow "cheated" because he could not imagine any of them could figure it out. I have no idea if the story is real (I grew up in Norway, so Abel was a popular subject for stories like this) - it was told to me in high school by a maths teacher after giving us the modified task of seeing if we could find any shortcuts to doing the sums, and seeing what we'd come up with. I found the formula quickly, but at that age that's nothing special, especially not when prompted to find an alternative solution. But the overall idea the teacher was trying to get us to understand was how to pause and think about how to decompose a problem rather than just picking the most obvious alternative, and learning to be "lazy" in the sense of relentlessly looking for an easier way to do things is a large part of what got me into software development..
karmakaze 7y ago
This reminds me of the von Neumann fly puzzle story: https://en.wikipedia.org/wiki/John_von_Neumann#Cognitive_abilities https://en.wikipedia.org/wiki/John_von_Neumann#Cognitive_abi...
- eesmith 7y ago
  Or swallow, if you think Wigner's account is more accurate than Halmos'.
- hnews_account_1 7y ago
  Yeah, Neumann was well into the nutty end of the genius spectrum. As in, his genius was so nuts that it's hard to believe he was real. Similar with "The strangest man to ever visit my lab" - Paul Dirac (that's a quote from Neils Bohr). At least Dirac appeared to be autistic enough from the description that I'm not surprised he was able to do what he did. It's a superior intelligence, but I'm guessing he was obsessive about shit as well.
- saagarjha 7y ago
  Of course, once you know how to sum infinite geometric series it's not too hard to come across the answer that way ;)
wenc 7y ago
I think most of us are impressed by computational parlor tricks (and indeed raw computational intelligence in general -- being able to process information and compute quickly and accurately), but for me, genius goes beyond that. Genius is about having rare and useful insights that the rest of us are incapable of, and that a computer is unable to easily replicate. For instance, there was this thing on Twitter recently about all percentages being reversible (7% of 50 is equal to 50% of 7, but the latter is easier to mentally calculate). Most of us are aware that multiplication is commutative, but it takes genius to recognize and frame that insight in a useful way.
- cheerlessbog 7y ago
  It never occurred to me that 7% of 50 is equal to 50% of 7 perhaps because they are equally easy to calculate? Multiply 7 by 5 and fix the decimal point.
  - wenc 7y ago
    What about 17% of 50? 17*5 isn’t so simple anymore, whereas 50% of 17 is 8.5. To be fair, with most shortcuts, it’s possible to construct difficult cases (17% of 23 is difficult in either order) but where it applies (when one of the pairs is a common percentage), exploiting commutativity can be quite useful. Plus the mental overhead of remembering the rule is extremely minimal.
    - JadeNB 7y ago
      How can you discover that 50% of 17 is 8.5 if you can’t multiply 17*5? (If the answer is “by halving”, then my response is that halving and then multiplying by 10 is often the easiest way to multiply by 5!)
      - Sharlin 7y ago
        I’m not sure, but to me at least halving 17 to get 8.5 is an almost completely intuitive process, whereas multiplying 17 by 5 seems to require an extra cognitive step to reduce it to shifting the decimal point and then halving (or vice versa). Never mind the extra step when presented by the problem of taking 17% of 50.
Psyladine 7y ago
Feynman was undoubtedly a genius, but he also suffered from a need to be admired. The safecracking episodes at Los Alamos are a perfect example - giving the impression he was an expert safe cracker when his real methodology was guesswork and sometimes subterfuge (birthdays, anniversaries, or even subtlety observing someone inputting their combination).
- elSidCampeador 7y ago
  I get your point, I really do. But if what you call his "real methodology" worked well, why would he use more advanced/safecracker-y techniques? I'd say the real mistake he made was that he lifted the veil off of how he did things, leading people to say "oh even I could have done that".
  - Psyladine 7y ago
    Something something, invoice $1 for chalk, $9999 for knowing where to mark the 'X'.
    - elSidCampeador 7y ago
      zigackly (https://farm5.static.flickr.com/4096/4746820383_314f398f2f.jpg https://farm5.static.flickr.com/4096/4746820383_314f398f2f.j...)
matt-noonan 7y ago
David Kelly [1] once told me that as a grad student at Princeton, he somehow managed to get the Sunday New York Times delivered to him late Saturday night. He'd stay up all night solving the crossword puzzle, then dazzle everybody who was stumped on it the next day. [1] https://www.vinc17.net/yp17/index.en.html https://www.vinc17.net/yp17/index.en.html

zamadatix 7y ago

I'm not sure I'd call it coincidence rather "being struck by the vast amount of time and passion Ramanujan put into mathematics". I'd be absolutely amazed if he couldn't have recalled a similarly obscure fact for most numbers below 10,000.

macewindu 7y ago

> Ramanujan's innate calculating capability. Hard work and obsessive work effort on a specific area makes it appear innate.

mkl 7y ago

> When you first hear the taxicab number story, your initial impression is to be struck by Ramanujan's innate calculating capability. This is the way the story is always presented, and I think that's usually how it's intended, but I think it's quite misleading for another reason too. If you've ever made or looked at a table of cubes, the famous fact really jumps out (in base 10). I'm serious, look: n n³ -------- 1 1 2 8 3 27 4 64 5 125 6 216 7 343 8 512 9 729 10 1000 11 1331 12 1728 The two pairs of cubes are 1000 and 729, and 1728 and 1, and 1000 and 1 make the addition trivial and the similarity obvious (and 729 and 1000 are even right next to each other, one row away from 1728!). With that observation, it doesn't take much effort to try the smaller possibilities and see that 1729 is the smallest number that can be written as the sum of two cubes two different ways. Ramanujan knew numbers and their relationships intimately, better than Hardy, who knew more theory. I think Ramanujan knew the fact about 1729 already, and that you are right about the taxi number coincidence being more surprising and, well, impressive. (Yes, I've commented on this before: https://news.ycombinator.com/item?id=21165031 https://news.ycombinator.com/item?id=21165031)