8 ms·
9,73,241,561,1081,1849,_?_
- perone 17y agohttp://www.wolframalpha.com/input/?i=9%2C73%2C241%2C561%2C1081%2C1849 http://www.wolframalpha.com/input/?i=9%2C73%2C241%2C561%2C10...
- markerdmann 17y agoWolfram Alpha has been a bit of a disappointment in terms of natural language parsing, but it's hard not to be impressed by the way it handles mathematical inputs. Does anyone know if the natural language parsing is showing improvement, or does it still choke on most inputs?
- wizard_2 17y agoI'm quite impressed with the differences table. 9 | | 73 | | 241 | | 561 | | 1081 | | 1849 | 64 | | 168 | | 320 | | 520 | | 768 | | | 104 | | 152 | | 200 | | 248 | | | | | 48 | | 48 | | 48 | | |
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- ComputerGuru 17y agoI read the title here which has no spaces between the numbers (a real pet peeve :P) then went on to read the explanation of "Edwin's" attempts to solve the blank. Except without the spaces, I thought he was trying to find the remaining digits of a single number (with commas for thousands separator). (before clicking the link, I was expecting "find the remaining digits to make this number a prime" or something) Nice link though!
- fsniper 17y agoI hit my head to wall after reading this. Difference table is a really cool way to solve this kind of questions. I thought "how come I have never learned this difference table before". I could make better points with IQ tests with difference tables :)
- megamark16 17y agoI've never been particularly good (or confident?) at math, but this actually piqued my interest, it's a very cool trick/technique. Maybe I should enroll in a math class at the local community college just to try to improve myself a little bit. Besides, I have so many more ways to actually apply what I'd be learning now then I did when I took my last math class (which I think was back when I was 16 or 17).
- rw 17y agoYou'll probably get farther with self-study. I suggest picking a field or two (discrete mathematics, number theory, graph theory, topology, and abstract algebra are probably the most useful and accessible ones to you right now) and then meeting with a professor a few times a month to go over homework problems. Doing it that way will prepare you to teach yourself the material, giving you the confidence you claim you don't yet have.
- hurt 17y agoI've also never heard of difference tables before, what a cool little technique!
- JoeAltmaier 17y agoI discovered difference tables as a kid, didn't know they were used much, I was modelling motion on a 2D display (an oscilloscope hooked to a P2P11!). Was a lifesaver for quick polynomial evolution - uses only addition, execution time scales linearly with the order of the polynomial. So, how do you go from the difference table "coefficients" to the polynomial?
- eru 17y agoIt's a bit non-trivial, but mechanically solvable. Polynomial coefficients are for normal powers. Difference table coefficients have more to do with falling powers (x * (x-1) * (x-2) * ...). Look up "Finite Calculus" or "Difference Equations" if you are interested.
- tudorachim 17y agoI'm not sure if this is your question, but divided differences are a way to do polynomial interpolation, with efficient updating to account for new data points: http://en.wikipedia.org/wiki/Newton_polynomial http://en.wikipedia.org/wiki/Newton_polynomial
- jonsen 17y agoIntegrate the constant 48 back to the series level: 48 -> 48x -> 24x^2 -> 8x^3 Subtract 8x^3 from the series 9 - 8•1^3, 73 - 8•2^3, 241 - 8•3^3 and repeat the whole process on this series for the coefficient of x^2, etc...
- SandB0x 17y agoExactly. The other perspective: When you see 'difference', think 'derivative' . Approximately, what the difference table is doing is differentiating until the leading term is constant. If we end up with a constant k levels down, we must have had a.x^k in the polynomial, as the derivative of a.x^k = ak.x^(k-1). Repeating this k times gives constant = a.k! So the leading coefficient of the polynomial, a = constant/k! I guess the next step is to subtract the values generated by this term from the sequence, then repeat to find the next term in the polynomial...
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- jtnak 17y agowhy is this at the top?
- iamwil 17y agobecause it is interesting.
- amichail 17y agoThis question is not well-defined. Any number can be a solution.
- bdr 17y agoHere's one interpretation: What will the simplest program that outputs these numbers output next?
- amichail 17y agoHow will you prove that your answer is correct?
- Leon 17y agooccam's razor
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- bdr 17y agoWell, that could be done by brute force. Realistically, though, it's more about whether anyone can beat your answer.
- jcl 17y agoI don't know why you're being modded down. To elaborate: You can define "simplest program", for instance, as "shortest string consisting of [0-9], plus, minus, times, parentheses, and the symbol 'n' that when evaluated as a mathematical expression yields the given sequence for n=1, 2, 3...". You can then find a suitable string, like "((8 * n + 4) * n - 4) * n + 1", and do an exhaustive search of shorter strings to show that no better string exists. That would be 16^16, or less than a year on a computer that can knock off a trillion strings per second -- less if you use logic to constrain the search space. However, realistically, a person asking such a question is merely looking for a "good" solution, not necessarily a proven-optimal one, as there may be multiple reasonable explanations for a given sequence. Rather, the value of the "simplest program" criteria is that it allows you to compare proposed solutions. So, yes, for any given number, there is a formula to make it next in the sequence. But by applying the "simplest program" criteria, we see that some formulas -- and hence some values -- are better than others.
- graywh 17y agoWas surprised to see this happen: http://www.wolframalpha.com/input/?i=0%2C1%2C2%2C4%2C8%2C16%2C32%2C64%2C128%2C256%2C512 http://www.wolframalpha.com/input/?i=0%2C1%2C2%2C4%2C8%2C16%...
- atamyrat 17y agoWhy start with 0? 1,2,4,8 works fine. http://www.wolframalpha.com/input/?i=1%2C2%2C4%2C8%2C16%2C32%2C64%2C128%2C256%2C512 http://www.wolframalpha.com/input/?i=1%2C2%2C4%2C8%2C16%2C32...
- lucifer 17y ago0, ...
- nixme 17y agoWorks properly if you take out the 0 term: http://www.wolframalpha.com/input/?i=1,2,4,8,16,32,64,128,256,512 http://www.wolframalpha.com/input/?i=1,2,4,8,16,32,64,128,25...
- seven 17y agoThis works with 0: http://www.wolframalpha.com/input/?i=0,1,2,4,8,16,32,64,128,256,512,1024 http://www.wolframalpha.com/input/?i=0,1,2,4,8,16,32,64,128,...
- jriddycuz 17y agoYes but in the recurrence relation, it said "for all n >= 1". I guess the additional term was the tipping point that let mathematica infer that the meat of the sequence was for n >= 1.
- jriddycuz 17y agoAgreed, don't start with zero for that. There's no real number n such that 2^n = 0, so the the closed-form expression you were seeking won't be found that way.
- moron4hire 17y agoNot to be too harsh here, but... I've been using "difference tables" (without calling it that) since I was 10 years old. I don't mean this to be bragging at all, because I didn't think (and still don't think) it was at all remarkable. It's just a basic method of analysis.
- Emore 17y agoSame here. We learned of them in 4th grade, in Sweden.
- Wilduck 17y agoLearned them in 5th grade "wings" class in Minnesota.
- pvandehaar 17y agoI home-schooled myself for a year and stumbled upon them, and decided to call them "differentials" because I had heard that term before. It's also possible I was had been sitting too close to my brother while he was studying them and had forgotten about it for a time.
- niyazpk 17y agoYou can see a discussion about difference tables here: http://www.physicsforums.com/showthread.php?t=114995 http://www.physicsforums.com/showthread.php?t=114995
- tkahn6 17y agoCan anyone explain why it takes so long to find the equation differentially? I understand that is subjective, but the 'constants' are only constant from y''' to y''. From y'' to y' the 'constant' is linear (meaning you have to solve for change in y' - y) and the 'constant' is quadratic from y' to y. Basically you can't just integrate each DE and solve for a constant C by finding the value of the lower order DE where x=1 and finding the difference. Why not? I'm in Calc BC in high school so this could be a stupid question.
- shrughes 17y agoHuh? What DE?
- jacobolus 17y agoThis is just a sequence being modeled by a polynomial. The point of a "difference table" is that when you take enough levels of differences (if you like you can think of these as analogous in some way to derivatives, insofar as each step of grabbing differences reduces the order of a polynomial sequence by one power, but it's not really necessary), you eventually get down to a constant, from which you can work backwards to figure out what the patterns in higher-up differences are.
- tkahn6 17y agoI'm aware difference tables can be used to find the final term in the sequence - it's what I did to find the answer. I was using differential equations to find the equation that describes the sequence and I was wondering why the constant that you get when you integrate a function isn't actually a constant after the second integration.
- IsaacSchlueter 17y agoPlease don't post your homework to the list.
- camccann 17y agoThis method of computing successive terms of a polynomial actually has an interesting place in the history of computers; the basic principle is the same as that used by Charles Babbage's difference engine. Wikipedia article on the difference engine: http://en.wikipedia.org/wiki/Difference_engine http://en.wikipedia.org/wiki/Difference_engine Just for fun, a difference engine built with Legos: http://acarol.woz.org/ http://acarol.woz.org/
- redorb 17y agoI loaded into excel, then copy and dragged down to see what excel would say is the next 2 answers ... I got the following ...1081,1849,[1890.067],[2248.467].... Now I just wonder how excel got these answers :/
- ajscherer 17y agoDrag down a few more cells and use a difference table on the resulting sequence.
- pfarrell 17y agoI learned about difference tables in Conway and Guy's great book "The Book of Numbers". I wish the book had been available when I was an undergrad. It describes everything I loved in math that I wasn't able to pinpoint. There are so many little gems in it, I've spent months reading it halfway through and then starting over to pick up what I missed. Highly recommended. http://www.amazon.com/Book-Numbers-John-H-Conway/dp/038797993X http://www.amazon.com/Book-Numbers-John-H-Conway/dp/03879799...
- l0nwlf 17y agoDifference table only works in case of arithmetico-geometric series or polynomial series. It fails in other sequences like fibonacci,prime etc. However if given a sequence I always try difference table first as there are high chances that it'll be cracked.
- leif 17y agooleis turns up nothing, sadly
- bd 17y agoIf somebody wants to try this method on different sequences, I made a simple solver in JavaScript: http://alteredqualia.com/visualization/hn/sequence/ http://alteredqualia.com/visualization/hn/sequence/
- osteele 17y agoThe wonderful (satirical, but mathematically sophisticated) book Mathematics Made Difficult introduces difference tables, and uses them to demonstrate how to compute the next number in the sequence [1,2,4,8,16,…] – that number being, of course, 31. This answer is correct in that it is the next item in the lowest-order polynomial that generates the first five terms. This reveals both the strength and weakness of difference tables (and the flavor of Mathematics Made Difficult).
- g__ 17y agoThis answer is also correct in another way :) http://threesixty360.wordpress.com/2008/04/30/1-2-4-8-what-comes-next/ http://threesixty360.wordpress.com/2008/04/30/1-2-4-8-what-c...
- jacobolus 17y agoHow does that work? Like this? That's not really following the "pattern" though, which is to keep adding 1 down the left edge of the triangle. Choosing to put a 0 in because we ran out of terms in our series is pretty arbitrary. 1 2 4 8 16 31 1 2 4 8 15 1 2 4 7 1 2 3 1 1 0
- req2 17y agoDifference tables don't operate on a "pattern"- they simply take the difference of the two terms above.
- jacobolus 17y agoNo, but the point is that using a difference table to find the "next term" in a sequence is rather stupid if the difference table doesn't terminate until you run out of terms.... the implication of that is that you have a sequence which cannot actually be fully described via that difference table.
- albertsun 17y agohttp://www2.research.att.com/~njas/sequences/ http://www2.research.att.com/~njas/sequences/ This is a great resource for finding information on integer sequences significant in combinatorics and other formal math topics. It didn't have the solution to this particular sequence, because, afaik there is nothing particularly interesting about it.
- psyklic 17y agoAlso of note is their Superseeker computer -- email it your sequence and it will perform a lot of additional analyses on the sequence beyond the normal web interface. http://www2.research.att.com/~njas/sequences/ol.html http://www2.research.att.com/~njas/sequences/ol.html
- roundsquare 17y agoIs there interest in people posting neat math puzzles on HN more frequently? There's a few I really like and I would love to see what puzzles others have.
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- sujaym 17y agoThe series 9,73,241,561,..corresponds to: 2^3+1^2; 4^3+3^2; 6^3+5^2; 8^3+7^2; 10^3+9^2; 12^3+11^2; ? Soln: 14^3+13^2 = 2913