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gavinksong

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  1. gavinksong's post in Equilateral Triangles: Horizontal = Slant was marked as the answer   
  2. gavinksong's post in 6 Squares and a Common Point was marked as the answer   
    I made a mistake. 'eight distinct' should be 'six distinct'.
  3. gavinksong's post in Reflect on this was marked as the answer   
  4. gavinksong's post in Waiting spider was marked as the answer   
  5. gavinksong's post in Nicholas and Peter share their nuts was marked as the answer   
  6. gavinksong's post in Find The Apples was marked as the answer   
    Since nobody has gotten this...

    Here's the answer:



  7. gavinksong's post in Covering perforated hexagon with triminoes was marked as the answer   
    Wait, I just realized that the question is asking about D(2015), not D(2014).




  8. gavinksong's post in Mirror Primes was marked as the answer   
    Congratulations! The answer I had in mind was sort of a mix between these two answers, but both of yours are valid as well.




  9. gavinksong's post in Six sixes to make 1,000 (modified version) was marked as the answer   
    Credit goes to all contributors for the solutions (including DejMar and plasmid).

    Thank you, PerhapsCheckItAgain, for this slick puzzle.
  10. gavinksong's post in Retirement was marked as the answer   
    This is incorrect because the minimum requirement only advances three months at a time. It does not suddenly jump forward by a year.



  11. gavinksong's post in Pecular Spectra was marked as the answer   
  12. gavinksong's post in Stand and topple was marked as the answer   
  13. gavinksong's post in Easy magical Triangle was marked as the answer   
  14. gavinksong's post in More fun with Fractals was marked as the answer   
  15. gavinksong's post in Domino Overhang was marked as the answer   
  16. gavinksong's post in Chase on an endless road, continued was marked as the answer   
  17. gavinksong's post in Chase on an endless road was marked as the answer   
  18. gavinksong's post in Chaos calculation was marked as the answer   
    I posted this in the original thread, but I think it belongs here now.


    Well, for a triangle, if we add up the white space, it is just the infinite sum of a geometric series with a factor of 3/4 (the big empty triangle in the middle is one-fourth of the total area, the three smaller triangles are one-fourth of the remaining area, and so forth). Strangely, it converges at the total area of the triangle. It we calculate the black space, it is three-fourths raised to the infinite power, which is zero. So apparently, the entire triangle is supposed to be prohibited space.

    I think here bonanova's musings becomes relevant:


    Instead of looking backwards, as we did to solve the problem initially, we should look forwards from the initial point. This is all intuition, but you will notice, that if the initial point is in the big empty triangle in the middle, then in the next step, it will always move to one of the three smaller white triangles. And from there, it will always move into one of its three smaller "shadows" (I think that is an appropriate term, and hopefully you understand what I mean), and so on. The important point here is that we always move into smaller and smaller "shadows".

    When the initial point is thrown into the triangle at random, the probability that it lies into a white triangle is 100%. However, as the simulation progresses, the points are eventually forced into smaller and smaller white triangles. In other words, the amount of "prohibited space" is not actually an absolute value, but simply increases with step number (in a sense). As in, there is no prohibited space when you throw in the first point, but the second point cannot be in the large white triangle, the third point cannot be in the large white triangle or any of its shadows, the fourth cannot be in the large white triangle or its shadows or any of its shadows' shadows, and so on.

    So to us, it seems as though there is some sort of absolute law because we see the larger empty white triangles, but in reality, it is only because we do not see the infinitesimally small white triangles with the dots in them. The entire process is just endless error-correction, moving toward an impossible figure, trailing behind an interesting asymptotic pattern in the process.
  19. gavinksong's post in five real numbers was marked as the answer   
  20. gavinksong's post in Averages part I was marked as the answer   
  21. gavinksong's post in etch-a-sketch math was marked as the answer   
  22. gavinksong's post in Stand behind your friend was marked as the answer   
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