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Mysterious number 6174
- raganwald 15y agoIt looks like this essay is saying that 6174 is a fixed point for Kaprekar's operation on four-digit numbers. http://en.wikipedia.org/wiki/Fixed_point_(mathematics) http://en.wikipedia.org/wiki/Fixed_point_(mathematics)
- pnathan 15y agoI am drawn to wonder if there is a discrete limit operation, and taking the discrete limit of Kaprekar's operation on all 4-digit numbers yields 6174. My mastery of fixed-point mathematics is rusty, so I am unsure if that is identical to the above...
- eru 15y agoPerhaps you are talking about http://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior#Special_case:_discrete_metric http://en.wikipedia.org/wiki/Limit_superior_and_limit_inferi... ?
- pnathan 15y agoSeems like it! Cool, thanks. (At work, so can't say definitely yes/no without studying).
- shou4577 15y agoIn fact, it says that 6174 is more than just a fixed point, it is also the only fixed point, and an attracting fixed point. That is, every four digit number with transform to 6174 in finite (indeed, less than 7) steps under Kaprekar's operation. This is significantly stronger than just being a fixed point. As an analogy, when modeling population dynamics, one often finds both stable and non-stable equilibrium solutions. The stable ones are the only solutions that can occur in nature, and so their stability is an important thing to note. This stability often (I hesitate to say always) is the result of the solution having the same attractive nature as 6174 does.
- lotharbot 15y agoAnother fixed point of the operation is zero, which is reached in a single step if you start with a number with all the same digits (1111, for example.) Zero is stable and attracting, it just has a very small basin of attraction.
- waterhouse 15y agoIf you want to see precisely which numbers lead to which fixed points, and in how many steps, I've posted it here for all 4-digit numbers: http://pastebin.com/3iXT777N http://pastebin.com/3iXT777N
- sesqu 15y agoHere's a plot: http://imageshack.us/f/192/stepsd.png/ http://imageshack.us/f/192/stepsd.png/ It might be interesting to plot the mapping instead. edit: one has already been made: http://news.ycombinator.com/item?id=2626879 http://news.ycombinator.com/item?id=2626879
- ableal 15y agoThanks, but there's a bug in your code/results - for example, 1011 does not converge to 0, it does hit 6174 in 5 steps. Only 1111, 2222, etc. yield 0. I suspect the handling of 3-digit values. I also slipped there at first ;-)
- waterhouse 15y agoFor 1011, the next step is 1110 - 0111 = 999, and then... apparently you're supposed to handle that as 9990 - 0999, rather than 999 - 999. Huh. I see. But I'm inclined to consider that a bug in the definition, rather than in the code. :-} The four-digit-ness came from applying the operation to four-digit numbers, and saying "oh yeah and we'll zero-extend the results to four digits" introduces an ugly element of redundancy. However, we can make it not ugly anymore by making "zero-extend everything to make it be 4 digits" the primary condition. Instead of applying it to 4-digit numbers, we'll make every number have 4 digits and then apply the Kaprekar procedure. So we'll now be working with the numbers 0000-9999, rather than 1000-9999. (There's no way to zero-extend numbers bigger than 9999 to be 4 digits, so that's all.) Here are the results for 0000-9999: http://pastebin.com/TQ3cMshV http://pastebin.com/TQ3cMshV I think the frequencies of the number of steps to equilibrium are somewhat nicer, as well, when you count 0000-0999. As a simple metric, they have more factors of small primes, especially 2. 1000-9999: 0 steps: 1 = 1 1 steps: 365 = 5 * 73 2 steps: 519 = 3 * 173 3 steps: 2124 = 2^2 * 3^2 * 59 4 steps: 1124 = 2^2 * 281 5 steps: 1379 = 7 * 197 6 steps: 1508 = 2^2 * 13 * 29 7 steps: 1980 = 2^2 * 3^2 * 5 * 11 0000-9999: 0 steps: 2 = 2 1 steps: 392 = 2^3 * 7^2 2 steps: 576 = 2^6 * 3^2 3 steps: 2400 = 2^5 * 3 * 5^2 4 steps: 1272 = 2^3 * 3 * 53 5 steps: 1518 = 2 * 3 * 11 * 23 6 steps: 1656 = 2^3 * 3^2 * 23 7 steps: 2184 = 2^3 * 3 * 7 * 13
- Ruudjah 15y agoEvery number is special. Every number has something on where you can apply rules to, and thus makes it special. Is there a database containing all the numbers and a list of atributes that make them special?
- andrewflnr 15y agoAll the numbers? ALL of them? :)
- GrangalanJr 15y agoYes. All 45,000,000,000 of them. :-) http://www.youtube.com/watch?v=drE5cHe6c3s http://www.youtube.com/watch?v=drE5cHe6c3s
- _delirium 15y agoThere's an old mathematics joke proving by contradiction that there cannot exist uninteresting integers: http://en.wikipedia.org/wiki/Interesting_number_paradox http://en.wikipedia.org/wiki/Interesting_number_paradox
- ZeSmith 15y agoWhich domain? Reals? That's a lot of numbers. Infinity, in fact.
- pavel_lishin 15y agoNot just an infinity, but a bigger infinity than the integers.
- tambourine_man 15y agoThat has always bothered me deeply. We measure something by comparing its frontiers (where it begins and ends) to something else. If this something has none, this clearly cannot be done. Of course, it's also clear that between 0 and 1 you have infinite real numbers. But is an infinite inside an infinite enough to provide a size hierarchy? Maybe the quantum physics guys will prove that the universe is granular in every possible level and that everything is just an enormous pile of huge natural numbers. Then infinity and paradoxes will just be a fun thought experiment, and reality will still be pragmatically ungraspable, but profoundly boring.
- andrewflnr 15y agoThis is indeed very interesting, but it's no surprise that it's just incidental. This and similar phenomena are just artifacts of our base-10 number system.
- clvv 15y agoSome related references: http://mathworld.wolfram.com/KaprekarRoutine.html http://mathworld.wolfram.com/KaprekarRoutine.html http://en.wikipedia.org/wiki/6174_(number) http://en.wikipedia.org/wiki/6174_(number) The Wolfram link contains a list of Kaprekar numbers and Kaprekar sequences in common bases.
- ChuckMcM 15y agoIts all fun and games until you think May 21st is the date the world will end. More seriously, numbers, and number theory, can be quite interesting and often leads to computational insight in algorithm computation or numerical solving. But ultimately, unless you're a numbers geek, it [ edit: Kaprekar numbers [1] ] doesn't [ don't ] really teach anything. [1]Sheesh, this place has lost its sense of humor.
- Xk 15y ago> Its all fun and games until you think May 21st is the date the world will end. ... what? That has nothing to do with number theory. > More seriously, numbers, and number theory, can be quite interesting and often leads to computational insight in algorithm computation or numerical solving. But ultimately, unless you're a numbers geek, it doesn't really teach anything. There are a number [1] [2] [3] [4] of practical uses of number theory that do teach something. I've only listed the simplest uses with respect to cryptography, because those are the ones with which I am familiar, but I assure you, number theory is one of the more applied branches of mathematics. Granted, some number theory has very few applications. But you won't ever find something practical if you're always asking "Is this practical? I better not look further if it isn't." [1] http://en.wikipedia.org/wiki/RSA http://en.wikipedia.org/wiki/RSA [2] http://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exchange http://en.wikipedia.org/wiki/Diffie%E2%80%93Hellman_key_exch... [3] http://en.wikipedia.org/wiki/Digital_Signature_Algorithm http://en.wikipedia.org/wiki/Digital_Signature_Algorithm [4] http://en.wikipedia.org/wiki/Elliptic_curve_cryptography http://en.wikipedia.org/wiki/Elliptic_curve_cryptography
- raganwald 15y agoLet's say you fancy yourself as a cryptographer. You come up with a black box, it's a function that takes a number and produces another number. You figure you can use it to build a pseudorandom stream of numbers that will be the source of a cipher. Your algorithm is to start with a random number (the key) and and feed it to your function, producing another number. Which you feed to your function, producing a third number. And so you go, generating a stream of numbers for your cipher. After reading how Kaprekar's operation leads to a fixed point for four digit numbers and various cycles for other numbers, you might be nudged into investigating to see if there are inputs for your function that lead to cycles and fixed points. Which would pretty-much break your cipher entirely. Is this interesting? Yes. Is there computational insight? There is for me. Does it teach anything? I dunno, how is teaching something distinguished from providing insight?
- 51Cards 15y agoThanks for posting this, really enjoyed reading it.
- thesz 15y agoIf you look close enough, all those "special" numbers are divisible by 9. I made a quick experiment with other bases. 3-digits numbers, base 8 has fixed point 374_8=252_10. 252 `mod` 7 = 0. base-9, base-11 and base 13 seem do not have fixed points. base-12, 3 digits has (at least one) fixed point 5b6_12 = 858_10. 858 `mod` 11 = 0. I think, it exploits some properties of positional notation of numbers so that fixed points all divisible by (base-1).
- davnola 15y agoNot just the special numbers. Kaprekar's operation always results in a multiple of 9. Given a > b > c > d. Kaprekar's operation gives: 1000a + 100b + 10c + d - (1000d + 100c + 10b + a) =999(a - d) + 90(b - c) =9(111(a - d) + 10(b - c)) And so on for integers with other numbers of digits.
- jimfl 15y agoAny number subtracted from a re-arrangement of the digits results in a multiple of 9.
- sesqu 15y agoWhy?
- jimfl 15y agoA number is the sum of its digits, raised to the appropriate powers Sum(i=0..n, a[i]*10^i) 123, for example is 3 + 2 * 10 + 1 * 100. A power of 10 (10^n) can be re written as (1 + 9 * Sum(i=0..n-1, 10^i). For example: 1000 = 1 + 999 = 1 + 900 + 90 + 9 When you subtract two numbers with the same digits, you end up being able to factor a nine out of these sums fairly easily. I wrote a proof of the number - reverse(number) a while back. It can be found on archive.org http://web.archive.org/web/20050314023901/http://jimfl.tensegrity.net/math/nines/ http://web.archive.org/web/20050314023901/http://jimfl.tense...
- alphaBetaGamma 15y agoI find Kaprekar's operation totally uninteresting because it is (or at least seems to me) totally linked to base ten representation of numbers. If at least you had a result like "in any base, the analogue of Kaprekar's operation in that base has a fixed point for digits of length 3 or 4 only", it would be mildly interesting. But as it is it's not math, just senseless symbol manipulation.
- JoachimSchipper 15y ago"Recreational mathematics". Don't take it too seriously, the practitioners don't either. But yes, the fact that this is bound to a decimal system suggests that there are no deep wisdoms here.
- alphaBetaGamma 15y agoI don't have anything against recreational math; I rather like it myself. But, IMHO, the fact that something is tied to the representation of numbers removes it from the realm of "math".
- william42 15y agoI'd say that the representation of numbers is an important part of math. That said, I'd agree that decimal is not a particularly interesting part of math. For reals, continued fractions are probably the most interesting representation, and have nicer mathematical properties, at that.
- _delirium 15y agoTo the extent that 'regular' mathematicians pay attention to these things (which a good number do), I don't think it's because decimal representations are inherently that interesting, but more because some of the number-theoretic and meta-mathematical questions are interesting and yet a bit puzzling. We don't yet really have a great handle on proof techniques and understanding of these kinds of problems: why should some simple-to-state claims about integers and iterated operations on them be true, and others false, and why are some seemingly true ones so hard to prove (like the Collatz conjecture)? The fact that this one is about rearranging digits in base-10 integers is just an intuitively-appealing way of stating simple properties, but the interesting thing isn't really about the base-10-ness imo.
- GrangalanJr 15y agoMade a visualization: x coordinate is the initial numbers, y coordinates are the "intermediate" numbers hit on the way to 6174. Diagonal is obviously the plot of the starting numbers. :-) http://i.imgur.com/DfxRB.png http://i.imgur.com/DfxRB.png
- vorg 15y agoHave you compared this visualization with the one for 495 and with the two digit numbers?
- GrangalanJr 15y agoThree digits: http://i.imgur.com/kQSGr.png http://i.imgur.com/kQSGr.png And two: http://i.imgur.com/OaynE.png http://i.imgur.com/OaynE.png
- mohsen 15y agoi appreciate posts that anyone can enjoy regardless of their profession
- caf 15y agoKaprekar's operation on binary numbers behaves in a much more predictable fashion - for example, under Kaprekar's operation on N digit binary numbers, 2^(N-1) - 1 is always a fixed point.
- akarambir 15y agohere are few more details of this magic number. This article was submitted to HN but didn't get attention. http://ennovates.com/engineering/6174-number/ http://ennovates.com/engineering/6174-number/
- Luyt 15y ago"The program, which was about 50 statements in Visual Basic, checked all of 8991 four digit numbers from 1000 to 9999..." Whoa, you need 50 statements in VB for that? I need 14 in Python: import collections def kaprekar(k): nr = 0 while True: if k == 6174: return nr small = int("".join(sorted(list("%04d" % k)))) large = int("".join(sorted(list("%04d" % k), reverse=True))) k = large - small nr += 1 freqs = collections.defaultdict(int) for i in range(1000, 10000): if not i % 1111: continue # skip 1111, 2222, etc... freqs[kaprekar(i)] += 1 for k,v in freqs.items(): print k, v Outputs: 0 1 1 356 2 519 3 2124 4 1124 5 1379 6 1508 7 1980
- Sandman 15y agoAnd somebody programming in some other language might need even less lines. What's your point?