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Three Puzzles Inspired by Ramanujan
- quantumhobbit 10y agoThe house is #204 out of 288 houses. Assuming the house itself isn't counted in either sum, which wasn't too clear by the question. I got this by the definition of triangle numbers and solving n^2 = m(m+1)/2 for integer solutions. n is the house and m is the total of all houses. As usual, I have no idea how Ramanujan did this with cont. fractions.
- rhizome 10y agoThanks for the spoiler.
- quantumhobbit 10y agoSorry about that. It isn't the full answer though.
- madcaptenor 10y agoAn exposition of how Ramanujan did it: http://www.johnderbyshire.com/Opinions/Diaries/Puzzles/2009-06.html http://www.johnderbyshire.com/Opinions/Diaries/Puzzles/2009-...
- SilasX 10y agoMind blown: The article made me realize that you can generate one of those trippy "continued fraction" equalities any time you have x on both sides of the equation, one side by itself. Golden ratio: x = 1 + 1/x -> keep re-plugging the right-hand side into the x on the RHS: x = 1 + 1/(1 + 1/(1 + 1/...
- Jtsummers 10y agoGenerating Functions and their applications may be of interest to you. https://en.wikipedia.org/wiki/Generating_function https://en.wikipedia.org/wiki/Generating_function A PDF on the topic: https://www.math.upenn.edu/~wilf/gfology2.pdf https://www.math.upenn.edu/~wilf/gfology2.pdf I was introduced to them in Concrete Mathematics by Graham, Knuth, and Patashnik, but then got to see their application in later statistics courses.
- pklausler 10y agoIsn't the golden ratio x = 1/(x-1) ?
- pklausler 10y agoIt occurs to me that a continued surd also works here: x = sqrt(1 + sqrt(1 + sqrt(1 + sqrt(1 + ...
- goldenkey 10y agoYou might also have your mind blown by this esoteric result: https://news.ycombinator.com/item?id=11976554 https://news.ycombinator.com/item?id=11976554
- SilasX 10y ago... actually, just mired in incomprehension.
- leni536 10y agoYou can generate many of them, not all of them converges though.
- javajosh 10y agoThe sum of 1..n is n(n+1)/2. A. The sum of 1..N is N(N+1)/2. B. The sum of n..N = B-A. We want solutions to A = B-A. Or, 1 = B-A/A I leave the rest as an exercise for the reader. :)
- spraak 10y agoI really enjoy Quanta Magazine's artwork featured on the articles!
- bogomipz 10y agoIts great to see Ramunujan getting a fair bit of attention these days. His story is really fascinating. I can recommend: https://www.amazon.com/Man-Who-Knew-Infinity-Ramanujan/dp/0671750615 https://www.amazon.com/Man-Who-Knew-Infinity-Ramanujan/dp/06... This was also made into a movie a couple of months ago, I haven't seen the movie but it should be on netflix/itunes by now since I don't think it had a wide theatrical release. http://www.imdb.com/title/tt0787524/ http://www.imdb.com/title/tt0787524/
- madcaptenor 10y agoNetflix currently says they're getting it in August 2016 (DVD; I don't know about streaming)