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  1.  
    "Invert and multiply!" he cried, as the registers creaked and the half-adders rolled and swung. "Invert and multiply!" But it was too late, for the accumulators were dry and the buffers overflowing.
  2.  
    Multiply two numbers, each of which is the sum of two perfect squares. The result will also be the sum of two perfect squares.

    Funky, hunh?
    Brahmaputra - 600 ish AD
    Gauss - 20 page proof
    Bhargava - tiny proof very recently.

    So don't try this at home.
  3.  
    For example

    22 + 32 = 13
    42 + 52 = 41

    13 * 41 = 533
    = 232 + 22
    = 222 + 72

    Magic innit
  4.  
    What's described above is a special case of a far grander fundamental structure.

    The most trivial generalisation was discovered way back by Brahmaputra, who found that, for all integer variables:

    if for any k > 0
    n0 = a2 + k*b2
    n1 = c2 + k*d2
    then
    n0*n1 = x2 + k*y2

    The example given previously is a special case k=1 of this more general rule.

    Gauss generalised on Brahmaputra, and Bhargava generalised on Gauss.
    https://www.jstor.org/stable/3597249
    •  
      CommentAuthoroak
    • CommentTimeMay 22nd 2019
     
  5.  
    Physics intuition to the rescue
    https://phys.org/news/2019-07-illusive-patterns-math-ideas-physics.html
    Rather beautiful
  6.  
    •  
      CommentAuthorAngus
    • CommentTimeAug 9th 2019
     
    8/2(2+2)=?
    • CommentAuthorAsterix
    • CommentTimeAug 9th 2019 edited
     
    Who the hell writes ambiguous math equations in linear form nowadays?

    Well, if one used any common programming language (FORTRAN, BASIC, etc.--but not APL, which has no precedence), the answer would be 16, evaluated thus:
    (2+2) = 4
    then, left to right:
    8/2 = 4
    4 x 4 = 16.
    On the other hand, if you were using the APL evaluation (right-to-left, no precedence)
    (2+2) = 4
    then
    4*2 = 8
    8/8 = 1
    There are some outliers in the literature, but they're not common.
  7.  
    Why I ignored it
    •  
      CommentAuthorAngus
    • CommentTimeAug 9th 2019
     
    Exactly. It seems to have caused some consternation in places 'trappers probably never go.
    • CommentAuthorAsterix
    • CommentTimeAug 9th 2019
     
    Reminds me of some of the trick C language questions that Dennis Ritchie would put up on Usenet. More often than not, the correct answer would be "I don't know".
  8.  
    I remember them. Kind of APL-like.
  9.  
    apl is how I found out I'm not a programmer
    • CommentAuthorAsterix
    • CommentTimeAug 9th 2019
     
    I had a co-worker who simply referred to it as "that damned Iverson language".
  10.  
    I did like FORTH though. Prolog - not so much. Ditto Lisp.
    •  
      CommentAuthorAngus
    • CommentTimeAug 9th 2019
     
    I loved APL and always will. Used it for many years.
    • CommentAuthorAsterix
    • CommentTimeAug 9th 2019 edited
     
    Provided you had the character set installed. There were variants that used plain old ASCII. That got ugly fast what with di- and trigraphs for symbols.

    For LISP, you only needed parentheses and letters.

    The lead designer for a vector supercomputer told me that his kids learned APL as their first programming language. Must have been a culture shock to discover BASIC and FORTRAN.
    •  
      CommentAuthorAngus
    • CommentTimeAug 9th 2019
     
    • CommentAuthorAsterix
    • CommentTimeAug 9th 2019 edited
     
    IBM 2741 was another popular choice

    But there was also the Canadian MCM/70 if you wanted a portable APL computer.

    The IBM 5100 also came with an APL option