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      CommentAuthorAngus
    • CommentTimeMar 7th 2019
     
    Holy cow! Multiplication is commutative! Whoda thunk?
  1.  
    Last night I was watching a documentary on Egypt (8000 - 50 BC)
    https://www.youtube.com/watch?v=KuUMe-43A3E
    and fell asleep somewhere in the descriptions of the pyramid builders. I awoke dreaming about the numbers 8,13 and 21 and the idea that the sum of the reciprocals of two numbers could equal the reciprocal of their sum.

    I will spare you the onerous task of telling me that this does not work for 8,13 and 21 - nor indeed for any pair of real numbers. But do any pair of numbers satisfy the given relation?

    The answer is in the affirmative, and involves the real numbers 1, 2 and sqrt(3).
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      CommentAuthoraber0der
    • CommentTimeMar 19th 2019
     
    I don't know.
  2.  
    Posted By: Andrew Palfreyman1, 2 and sqrt(3)


    ...reminds me of a girl I knew in high school...
  3.  
    Reminds me of the damndest thing. The three numbers I cited form a right angled triangle, and the two solutions (there are two) form a pyramid shape. How curious.
  4.  
    https://www.quantamagazine.org/mathematicians-discover-prime-conspiracy-20160313

    "Soundararajan was drawn to study consecutive primes after hearing a lecture at Stanford by the mathematician
    Tadashi Tokieda, of the University of Cambridge, in which he mentioned a counterintuitive property of coin-tossing: If Alice tosses a coin until she sees a head followed by a tail, and Bob tosses a coin until he sees two heads in a row, then on average, Alice will require four tosses while Bob will require six tosses (try this at home!), even though head-tail and head-head have an equal chance of appearing after two coin tosses."

    Mind blown

    Explanation
    https://terrytao.wordpress.com/tag/kannan-soundararajan/
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      CommentAuthorTrim
    • CommentTimeApr 9th 2019
     
    Mathematicians Just Discovered an 'Astonishing' New Way to Multiply Large Numbers.

    https://www.sciencealert.com/mathematicians-just-discovered-an-astonishing-new-way-to-multiply-numbers-together
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      CommentAuthorE-Man
    • CommentTimeApr 9th 2019 edited
     
    Posted By: Andrew Palfreymanhttps://www.quantamagazine.org/mathematicians-discover-prime-conspiracy-20160313

    "Soundararajan was drawn to study consecutive primes after hearing a lecture at Stanford by the mathematician
    Tadashi Tokieda, of the University of Cambridge, in which he mentioned a counterintuitive property of coin-tossing:If Alice tosses a coin until she sees a head followed by a tail, and Bob tosses a coin until he sees two heads in a row, then on average, Alice will require four tosses while Bob will require six tosses (try this at home!), even though head-tail and head-head have an equal chance of appearing after two coin tosses."

    I think Quantum is kind of misrepresenting the problem. The completion constraints on Bob and Alice aren't identical. Which is why the probabilities are different. At least as far as I understand the problem.
    • CommentAuthorAsterix
    • CommentTimeApr 9th 2019
     
    Posted By: TrimMathematicians Just Discovered an 'Astonishing' New Way to Multiply Large Numbers.

    https://www.sciencealert.com/mathematicians-just-discovered-an-astonishing-new-way-to-multiply-numbers-together


    Wake me (and the rest of the world) when an astonishing fast way is found to divide large numbers. Trapdoor cryptology will become a thing of the past.
  5.  
    Posted By: E-Man
    Posted By: Andrew Palfreymanhttps://www.quantamagazine.org/mathematicians-discover-prime-conspiracy-20160313

    "Soundararajan was drawn to study consecutive primes after hearing a lecture at Stanford by the mathematician
    Tadashi Tokieda, of the University of Cambridge, in which he mentioned a counterintuitive property of coin-tossing:If Alice tosses a coin until she sees a head followed by a tail, and Bob tosses a coin until he sees two heads in a row, then on average, Alice will require four tosses while Bob will require six tosses (try this at home!), even though head-tail and head-head have an equal chance of appearing after two coin tosses."

    I think Quantum is kind of misrepresenting the problem. The completion constraints on Bob and Alice aren't identical. Which is why the probabilities are different. At least as far as I understand the problem.

    Care to elaborate on "completion constraints"?
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      CommentAuthoraber0der
    • CommentTimeApr 9th 2019
     
    OTOH, everything is quantum.
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      CommentAuthorE-Man
    • CommentTimeApr 9th 2019 edited
     
    Posted By: Andrew PalfreymanCare to elaborate on "completion constraints"?

    They make it sound like Alice is searching for HT and Bob is searching for HH. What I think Tadashi was on about was more about was Bob searching for HH and Alice searching for H with a T following at some point.

    Those are different constraints. So the idea that the likelihood of them is different isn't surprising. IMHO.
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      CommentAuthorDuracell
    • CommentTimeApr 9th 2019
     
    If the only options are H and T, then how would searching for HT be any different to searching for H with a T following at some point? Isn’t that merely similar to saying that Bob is searching for a H that is not followed by another H at some point?
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      CommentAuthoralsetalokin
    • CommentTimeApr 9th 2019 edited
     
    So you could take the binary expansion of pi as a string of random zeros (heads) and ones (tails) and search for instances of 0,1 and 0,0 and see how many of each there are in a certain slice of pi.

    11.001001000011111101101010100010001000010110100011000010001101001100010011000110011000101000101110000000110111000001110011010001001010010000001...

    I count 35 instances of 0,1 and 31 instances of 0,0 (without duplications) in that slice.
    Therefore... something is wrong somewhere.

    (Actually I suppose you should take two different but equal-length slices of pi, one for Bob and one for Alice, since they are each flipping their own coins.)
    •  
      CommentAuthorAngus
    • CommentTimeApr 9th 2019
     
    Clever.
  6.  
    At last a way to win at roulette. Simply bet on the opposite color from the last roll. Since two blacks, or two reds, come up less often on average than a black followed by a red, or a red followed by a black... you will slowly break the bank, in the long run.

    No?
  7.  
    • CommentAuthorAsterix
    • CommentTimeApr 12th 2019
     
    As I observed in another thread--figure out a comparable high-speed divide algorithm and you'll really have something.
    •  
      CommentAuthorAngus
    • CommentTimeApr 19th 2019
     
    Meanwhile in the master's common room. Sigismund arbuthnot the mad maths master musters his rhomboids.
  8.