9 ms·
6174
- deleted 3y ago[deleted]
- playingalong 3y agoThat's surprising. Any informal thoughts why would even a single 4-digit constant exist with this property? The intuition would be there are multiple cycles in this graph.
- Arnavion 3y agoThe first reference in the article has the working out. https://plus.maths.org/content/mysterious-number-6174 https://plus.maths.org/content/mysterious-number-6174
- dcow 3y agoThe conclusion is that we don’t know and therefore it might be coincidental. Hardly satisfying. It does appear there are cycles for other lengths.
- layer8 3y agoWhen does a mathematical fact become a coincidence?
- v64 3y agoRelevant reading: What Are Mathematical Coincidences (and Why Does It Matter)? by Marc Lange [1] [1] https://philosophy.unc.edu/wp-content/uploads/sites/122/2013/10/mathcoincMINDpublished.pdf https://philosophy.unc.edu/wp-content/uploads/sites/122/2013...
- cubefox 3y agoThe term "relevant" might be an understatement here.
- noah_buddy 3y agoLayman’s guess: if there’s no generalizable proof describing why this works and the only method of proof is enunerating every case. That said, I kinda doubt that that is the case here, we probably just need a craft way of describing this so that we may come up with a crafty way of proving it :)
- SkyBelow 3y agoIf there is a proof of no generalizable proof, that is interesting in and of itself. If there is not, then doesn't that mean we haven't found it yet (or proven it impossible, or proven it impossible to prove if impossible...). If we are still at the point of enumerating each case, that means our knowledge of the subject matter is lacking the deeper understanding to solve it in a nicer fashion.
- AnotherGoodName 3y agoNot an answer but a very good clue I can see is that the sum of the digits on each side of the equation consistently gets closer and closer. 9541 – 1459 = 8082 Left hand digit sum = 19. Right = 18. They are 1 apart. 8820 – 0288 = 8532 Both sides now = 18. Now 0 apart and they'll stay there. They are only 0 apart when at 18. 8532 – 2358 = 6174 Both sides = 18 7641 – 1467 = 6174 Both sides = 18 You can play with this a bit and it's consistent. The sum of digits of the left and right hand side consistency get closer to each other iteratively (but not necessarily closer to 18). Eventually they lock in at being equal to each other when their digits sum to 18. This seems to be one property to look at. I think there's then a second thing happening. Once the values on both sides have digits that sum to 18 the process from there converges on to 6174. So first the digits of the two sides to the equation converge to equal the same which always only occurs when the digit sum is 18. The digit sum locks into being at 18 at that point. And then subsequently once the digits are 18 they converge on to 6174. I would start by working out why digits on each side of the equation converge to summing to 18 on both sides of the equation and never being equal at any other value in this process. It reminds me of https://math.stackexchange.com/questions/99725/every-integer-is-congruent-to-the-sum-of-its-digits-mod-9 https://math.stackexchange.com/questions/99725/every-integer... Now the next thing I would do is ask why does every number with digits that sum to 18 eventually end up at 6174. 4 digit numbers with digits that sum to 18 is a very limited set so it should be easy to figure out the combinations and why they all reach 6174. Put those two together and you'd have an answer. (I'm thinking about it now but it really doesn't seem too hard).
- vanderZwan 3y agoThe follow-up article appears to give some hints though: > A number of readers emailed to say they had discovered that repeatedly adding up the digits of any of the kernels of Kaprekar's operation always equalled 9 (...) Professor Nishiyama has provided an explanation why this happens: it is because the result of performing Kaprekar's operation on any number is a multiple of 9. https://plus.maths.org/content/pluschat-15 https://plus.maths.org/content/pluschat-15 This is starting to look very similar to "if you repeatedly add all the digits of an integer represented in base 10 representation until you have a single digit, and the result is 3, 6 or 9, then it is divisible by 3". I forgot the exact explanation for that one, but IIRC has to do with the implicit calculation that is embedded in base 10 positional notation, other bases have a different number you can quickly verify the divisibility of this way. So maybe that (the "implicit calculation in base 10 representation" thing) is one part of the explanation. I mean whatever it is, it feels like a mix of all these operations imposing constraints upon each other and interacting with the recursive feedback loop to result in the convergence as an emergent property.
- Arnavion 3y ago>This is starting to look very similar to "if you repeatedly add all the digits of an integer represented in base 10 representation until you have a single digit, and the result is 3, 6 or 9, then it is divisible by 3". I forgot the exact explanation for that one, but IIRC has to do with the implicit calculation that is embedded in base 10 positional notation Yes. The number "xyz" is 100x + 10y + z. Each power of 10 can be split into 1 plus a multiple of 9, ie (x + y + z) + (99x + 9y). The second group where all the components are a multiple of 9 is of course divisible by 9 and by 3. The first group is the sum of the digits. So if the sum of the digits is divisible by 3, then the original number was divisible by 3. If the sum of the digits is also divisible by 9, then the original number was also divisible by 9. If the sum of the digits is not divisible by 3, the original number was not divisible by 3 either. This generalizes to any number of digits, and to any base N for testing divisibility by N-1 or factors of N-1.
- t-3 3y agoCheck out https://en.wikipedia.org/wiki/Kaprekar%27s_routine https://en.wikipedia.org/wiki/Kaprekar%27s_routine At a glance, there seem to be some patterns, like how for those bases with a 2-digit Kaprekar number the sum of the digits is base-1. There must be some number theory explanation for it.
- bmacho 3y agoThe graph is small. Similar graphs (e.g. this algorithm for 2 or 5 digit, or 3-4 digit numbers for base 9 or 11) have multiple cycles: https://en.wikipedia.org/wiki/Kaprekar's_routine#Kaprekar's_constants_and_cycles_of_the_Kaprekar_mapping_for_specific_base_b https://en.wikipedia.org/wiki/Kaprekar's_routine#Kaprekar's_...
- penteract 3y agoOne thing that makes it less surprising is that there are lots of numbers which map to the same result - for example all permutation of a bag of digits. I checked, and there are only 55 distinct results (54 excluding 0000) from applying the process to all 4-digit numbers, which leaves less space for lots of cycles.
- underlipton 3y agoDoes it work for different base systems? As someone who failed high school calculus (i.e., a moron), I would imagine that it's an artifact of base-10.
- codeflo 3y agoThere’s so much numerology in the world, even among smart people, that I think this is worth being pedantic about: There’s no such thing as a “four-digit number”, only a four-digit base-10 numeral. And facts about base-10 numerals aren’t facts about numbers.
- recursive 3y agoA lot of numbers have representations in base-10. A fact about the base-10 digits is a fact about the base-10 representation of the number, which is also a fact about the number. You might be able to satisfy yourself by replacing "the digits of" with "the decimal digits in the base-10 representation of".
- codeflo 3y agoThe point is that most of the time when digits are mentioned, it’s only a coincidental fact about one inelegant representation of the number — and often people are insufficiently aware of that.
- bee_rider 3y agoBase-10 isn’t inelegant, is it? I mean there are good arguments for 12 being better but it isn’t like 10 is prime or anything. Happened across a neat comment yesterday that presents a defense of ten. Not 100% convinced but it is interesting to see pushback. https://news.ycombinator.com/item?id=39000882 https://news.ycombinator.com/item?id=39000882
- dr_dshiv 3y ago1+2+3+4=10 And you can swear by that, if you know what I mean. * * * * * * * * * *
- spenczar5 3y agoRational fractions will terminate only if the denominator’s prime factors are the base’s factors. So for example, 1/2 = 0.5 and 1/5 = 0.2, but 1/3 = 0.333… and 1/7 = 0.142857…. 1/4 = 0.25 works because the prime factors of 4 are 2 and 2… but 1/6 fails because 6 is infected by 3. Now, base 12 has 2 prime factors (2 and 3) so it much any better than 10 really. But may I introduce base 30 (235)? Or perhaps base 210 will strike your fancy?
- dang 3y agoRelated: Mysterious number 6174 - https://news.ycombinator.com/item?id=2625832 https://news.ycombinator.com/item?id=2625832 - June 2011 (64 comments) 6174 - https://news.ycombinator.com/item?id=1625606 https://news.ycombinator.com/item?id=1625606 - Aug 2010 (1 comment) Mysterious number 6174 - https://news.ycombinator.com/item?id=480200 https://news.ycombinator.com/item?id=480200 - Feb 2009 (41 comments)
- Ontol 3y agoit is y combinator
- clktmr 3y agoArticle says it does work for all numbers except repdigits, but I think it fails for all palindromic numbers?
- mysterydip 3y ago1221 would become 2211, which avoids the difference becoming zero.
- fekunde 3y ago2211 1122 1089; 9801 1089 8712; 8721 1278 7443; 7443 3447 3996; 9963 3699 6264; 6642 2466 4176; 7641 1467 6174
- vikingerik 3y agoThis process doesn't just reverse the digits (leaving a palindrome unchanged) - first you sort the digits (which does change a palindrome), then reverse them.
- chasing 3y agoA rabbit hole into poking around a whole mess of Wikipedia pages about specific numbers, which was pretty entertaining. That said, https://en.wikipedia.org/wiki/List_of_numbers https://en.wikipedia.org/wiki/List_of_numbers is woefully incomplete.
- deleted 3y ago[deleted]
- jlv2 3y agoYes, the List of numbers page is inadequate. When I give a monetary gift, I like to make it start with an interesting number. And then I ask the recipient (e.g., my kids, or neices/nephews, etc) if they can figure out what the number is. e.g., 986.96 is based upon π sqaured.
- ximm 3y agoIn university I learned that every integer is interesting. Proof: If there are non-interesting integers, there must be a smallest non-interesting integer, which is an interesting property. ■
- xcv123 3y agoYour proof assumes there is only a finite set of non-interesting integers, but it could be an infinite set.
- planede 3y agoEven if the set is infinite, there surely is a smallest non-negative interesting integer.
- chasing 3y agohttps://en.wikipedia.org/wiki/Unexpected_hanging_paradox https://en.wikipedia.org/wiki/Unexpected_hanging_paradox
- prvc 3y agoWhy do you find this to be significant?
- waynesonfire 3y agoCome back when you ask that about yourself.
- code_runner 3y agoI don't understand the significance of this at all other than its the coolest thing I've seen on HN in a while. I couldn't be farther from a math nerd.... I avoided it as much as I could throughout school.... but things like that are just so interesting and weird. How on earth (and for what reason) did they find this out? The properties of this number are interesting enough but the process to discover it is just so crazy.
- LambdaComplex 3y agoGo read Lockhart's Lament. Maybe you actually do like math, but school is just terrible at teaching it.
- SoftTalker 3y agoI hated arithmetic (still do), but liked algebra, geometry, and calculus (forgotten most of it).
- JadeNB 3y ago> I hated arithmetic … That is to say, ambition, distraction, uglification, and derision.
- apitman 3y agoI really enjoyed the Grapes of Math by Alex Bellos
- ughitsaaron 3y agoI have a similar background with math as the commenter. +1 to Lockhart. If the commenter finds numbers interesting they may also really enjoy his book “Arithmatic.” I found it so refreshing. It truly reoriented my whole relationship with mathematics.
- subtra3t 3y agoAlso read Measurement, by the same author.
- 3y ago
- alex_young 3y agoThis seems obviously related to the standard manual accounting trick employed when isolating an error in a double ledger - the first thing you do is look to see if the error is evenly divisible by 9. If it is, you've transposed 2 or more numbers somewhere. To prove why this is so: For any numbers x and y: The correct value is 10x + y The transposed value is x + 10y The difference is (10x - x) + (y - 10y) Reduces to 9x - 9y Factors to 9(x - y)
- AnotherGoodName 3y agoYes i think there's two components to this and that's the first part. Each digit sum on the left and right side converges to 18. Each side of the equation a - rev_a = b has digit sums to that iteratively get closer to each other (sometimes stay the same distance but never getting further). Additionally that convergence only happens at 18. Eg. 5200 (sums to 7) - 0025 = 5175 (sums to 18, 11 apart) 7551 (sums to 18) - 1557 = 5994 (sums to 27, 9 apart) 9954 (sums to 27) - 4599 = 5355 (sums to 18, 9 apart) 5553 (sums to 18) - 3555 = 1998 (sums to 27, 9 apart) 9981 (sums to 27) - 1899 = 8082 (sums to 18, 9 apart) 8820 (sums to 18) - 0288 = 8532 (sums to 18, 0 apart) 8532 (sums to 18) - 2358 = 6174 (sums to 18, 0 apart) 7641 (sums to 18) - 1467 = 6174 (sums to 18, 0 apart) I think this is the first clue. The digits can only be equal on each side when they are 18 and the sum of each side progressively gets closer on each side, eventually equalling each other which has to happen at 18. I think if you dive in it's a variation of the classic 'digits sum to 0 mod 9'. Then once the digits on each side sum to 18 i think they must converge onto 6174 from there. So first we have digits always converging to have the same digit sum on each side and that convergence is always when the digit sum is 18 on each side of the equation. I think property is going to be provable by the classic mod 9 rules but it'd take some work. Then i believe we have a second property kicking in that all 4 digit numbers that have digits that sum to 18 on both sides of this equation will converge on 6174. This is a more limited set of numbers. Only numbers of the form a - a_rev = b that have digits that sum to 18 for both a and b need to be considered since we can separately see the convergence to 18 on both sides above.
- devilsAdv0cate 3y ago
- JKCalhoun 3y agoReminds me of some cylindrical contraption I saw at the Exploratorium in San Francisco over a decade ago. I believe too it was described even earlier in a "Scientific American" column — either Mathematical Recreations or Computer Recreations. It was some kind of device where a large horizontal cylinder was perhaps covered with numbers? Maybe there were rings or some other kind of "cursor" on the contraption? And I think as you rotated it there was some kind of math performed and, like this "6174" thing, it would seem to converge on a single number after so many iterations regardless of the starting state. Wish I could remember what that was.
- anigbrowl 3y agoI know the exhibit you're talking about; it's still there as far as I know. I haven't visited in about a year.
- pbhjpbhj 3y ago"The device you're referring to is likely the "Kaprekar Machine" at the Exploratorium in San Francisco. It's an interactive exhibit demonstrating Kaprekar's Routine and the convergence to the number 6174 through mathematical operations on a four-digit number." (ChatGPT 3.5)
- tetris11 3y agoProof? Hallucinated?
- pbhjpbhj 3y agoClosest thing I found with some manual searching was a blog that included an article on Kaprekar and a separate article on Exploratorium. The person seemed to be involved at some level so it suggested it was feasible, rather than providing evidence. Maybe archive.org would have something. Possibly hallucinaited (sic).
- brianpan 3y agoIn a year, ChatGPT 5 will point to this thread as proof.
- jaybrendansmith 3y agoFor 5, it's 98532. Fun stuff.
- eps 3y agoThere is no 6174-like number for 5 digits, only cycles and yours is a part of one of them: 74943 -> 62964 -> 71973 -> 83952 -> repeat 63954 -> 61974 -> 82962 -> 75933 -> repeat 53955 -> 59994 -> repeat https://kaprekar.sourceforge.net/output/sample.php https://kaprekar.sourceforge.net/output/sample.php
- esafak 3y agoAnother interesting four digit number associated with an Indian mathematician is 1729: https://en.wikipedia.org/wiki/1729_(number) https://en.wikipedia.org/wiki/1729_(number)
- ponsfrilus 3y agoThanks, nice one too!
- blt 3y agoI tried and failed to find the original paper by Kaprekar, does anyone have a pdf?
- botanical 3y agoYou're right, I can't seem to find it either. Tangentially, how much other research gets lost in the ether because it wasn't as interesting as this.
- blt 3y agoIt's probably on microfiche in a university library somewhere. HN elders: How long would it have taken to get your hands on this paper (or a similar "old; noteworthy but not famous" paper) in, say, 1985?
- quickthrower2 3y ago6174 isn't super special. Or should I say, isn't canonical. It is special but for base 10. But there are other numbers like this for other bases, and of course number lengths.
- alain94040 3y ago6174 is only remarkable if you count in base 10. This is HackerNews, so we all use hexadecimal. Sadly, according to https://kaprekar.sourceforge.net/output/sample_hex.php https://kaprekar.sourceforge.net/output/sample_hex.php, there is not a simple Kaprekar Constant in base 16.