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Mysterious number 6174
- jerf 18y agoWhat would an "explanation" be? An explanation with lower Kolmogorov complexity than the operation itself. That system of equations is probably all the "reason" you're going to get, since it's hard to get simpler than that. If you're looking for something more humanly profound, how about this: "Arbitrary operations on integers can yield arbitrary results." That's actually profound (in the Zen-like way a tautology can still be profound) in the context of "human understanding" and how people argue with numbers, but it would take a long time to explain why, if you don't already know.
- euroclydon 18y agoCall me naive, but I have always doubted the mathematics significance of rearranging the digits of an integer. When you rearrange 1234 to be 3241, you have lost all meaning. What real mathematical operation involves rearranging digits in an integer?
- fiaz 18y agoWho ever thought that x^3 transformed into 2x^2 would be useful one day? Of course I'm being facetious in this example, but the simple rearrangement of the mathematical expression above has profound implications for how we interact with the world. Yes I know that there is a whole derivation to the simple derivative example I've given, but who is to say that the 6174 transformation might have preceded an actual use? Who's to say that this odd mathematical transformation actually leads to something useful one day?
- euroclydon 18y agoI don't really see the connection. Your example is the derivative of a function, and what I am saying is that rearranging 1234 to 3421 doesn't mean anything to me. Now you have some new number who's only relationship to the old number is that it's one of (432)-1 possible permutations of it. Other than that, what's the significance?
- naveensundar 18y agoJust because there is no name for this operation does not mean it is not valid (or meaningful). There are many mathematical constructs which do not have any physical meaning.
- euroclydon 18y agoI don't really see the connection. Your example is the derivative of a function, and what I am saying is that rearranging 1234 to 3421 doesn't mean anything to me. Now you have some new number who's only relationship to the old number is that it's one of (432)-1 possible permutations of it. Other than that, what's the significance?
- imgabe 18y agoThe numbers in the article aren't rearranged arbitrarily, they're sorted to ascending and descending order.
- eds 18y agoDid you mean 3x^2?
- juanpablo 18y ago6174 it's the only real solution of this system of equations: 9 ≥ a ≥ b ≥ c ≥ d ≥ 0 abcd - dcba = ABCD
- Retric 18y agoAre you positive that's true for all bases?
- jerf 18y agoThe statement as juanpaulo gives it has no hidden dependence on bases. It is all spelled out by the conditions. The symbols a b c d stand for any values, not the "actual digits" 0-9, which are also merely placeholders for values with no "true meaning" of their own.
- Retric 18y agoHe is using both digits and the nubers that sequence of digits represents in that equation. In base 10, 7641 - 1467 = 6174. In base 11, 7641 - 1467 = 4808. Finding tricks that work in base 10 is interesting, but it's a much more fun to look for patterns that work across several number systems. A simple example is for any base > 3, (base - 1) times x where base > x > 1; you get a 2 digit number [x - 1],[base - x]
- jerf 18y agoNo, they are not. The symbols in the equation are simply numbers, and "9 ≥ a ≥ b ≥ c ≥ d ≥ 0" constrains them to 0-9. You are mistaking the map for the territory, sort of. The positional aspects of the number system get encoded the equations and then the mere "numbers" don't matter anymore.
- jmathes 18y agoYes they are. Where he meant 1000a+100b+10c+d he wrote abcd.
- 18y ago
- tel 18y agoA little conceptual snippet from GEB: Everything in mathematics can be seen from a "meaningless" perspective. In some sense everything from algebra to complex analysis can be stripped of meaning by thinking of them as just "a rearrangement of symbols according to a set of arbitrary rules". And, somewhere, that's really all they are. It's only once we apply an interpretation to these operations that they become "math". Same with numerical rearrangement. Just because it seems like it's just a algorithmic manipulation doesn't mean there isn't an interesting interpretation of the meaning somewhere. To be pathological, take the phrase X2YZ to mean "X + Y = Z if X < Y and Y - X = Z if Y > X" and both 1234 and 3241 are true. Now that manipulation suddenly has a whole lot of (silly) meaning.
- jmathes 18y agoOkay I'm only replying to this one reply but my point is relevant to all discussion re meaning. Always be suspicious of any statement about numbers that relies on the base. This property of 6174 is really a property of the number's representation in base 10. In another base, it's the same number, but doesn't have the same property. That's why it seems suspiciously meaningless.
- tel 18y agoOh, sure. My reply was more of a meta argument than really a defense of the meaning of 6174. There's no reason to say that there isn't some property of its representation in base N that isn't useful, but it's pretty unlikely to be general in any interpretation we're used to dealing with (thus the re-representation of "2").
- jacquesm 18y agoA nice example is the divisibility of numbers, by rearranging the digits and performing simple operations on them you can quickly figure out if a number is divisible by a certain other number without actually doing the division. for an example see: http://en.wikipedia.org/wiki/International_Bank_Account_Number http://en.wikipedia.org/wiki/International_Bank_Account_Numb... or google for 'eleven check'. A snippet from a yahoo answers page: "a number is divisible by 11 if and only if when you add the digits that are an odd number of places from the right, and then you add the digits which are an even number of places from the right, the difference of these is divisible by 11."
- huhtenberg 18y agoInteresting. Feels like a more mysterious version of Collatz conjecture :) which is - start with a positive integer, iterate as such: x = (x & 1) ? (3*x + 1) : (x/2); and you will eventually end up at 1. http://en.wikipedia.org/wiki/Collatz_conjecture http://en.wikipedia.org/wiki/Collatz_conjecture
- jdoliner 18y agoIt's not so much a question of mathematical use. In mathematics ideas are normally judged for implications for the rest of the field. This leads to a self perpetuating notion of what's important (things are important simply because they are). The notion of rearranging the digits of an integer has absolutely no mathematical significance. Because we represent numbers by strings of digits it's a very intuitive idea but I can promise you it will never lead to "significant results" as pe the current academic notion of mathematics. Furthermore mathematics has always been a search for generality. The more general a concept is the more cases to which it can be applied and normally the more impressive it is the mathematical community. I personally find this substantially more reasonable. Rearranging the digits of a number isn't even general to the concept of integers. It's dependent on the fact that we write our numbers in a base of ten. If this theorem held in other bases (like binary numbers) then that would be kind of cool. On another note the author uses the word kernel when I think he wants "fixed point". Kernel doesn't really have a meaning in this context.
- Shamiq 18y agoTo jdoliner: Please add some information to your profile so I can place your comment in some kind of reference. I love that you write clearly and to my untrained, therefore naive, eyes your arguments seem valid. But without an understanding of where you are coming from, I do not want to necessarily accept what you say on face value. I mean no insult, I just want more information. Cordially, Shamiq
- diN0bot 18y agoI agree, or add a note about whether you're a mathematician. I personally found this submission to be very exciting and interesting. I guess even with calculus, linear algebra and computability theory under my belt I still find number representation interesting. I don't think I'm alone. I'll use this idea later when tutoring kids in an attempt to share mathematical and programming excitement (would make for a fun script). This submission may not be significant in advanced mathematical theory, but getting distracted for 20 minutes doing arithmetic or programming is no less important, albeit at quite a personal level.
- 18y ago
- daveparker 18y ago"...it also has one more surprise up it's sleeve..." Fail.
- fh 18y agoDownvoted for implying there's anything mathematically special or interesting about base 10. Okay, I can't really downvote, but you get the idea.
- jacquesm 18y agofirst off, this is not about 'base 10', it is about interesting patterns found in series of digits, similar (but different) interesting patterns can be found in series of digits in other bases. Secondly, if every time a mathematically oriented person stumbles upon something interesting instead of figuring out why they would respond like you just did mathematics would grind to a halt really quickly. After all, that's how math started - that curiosity about patterns - and mathematicians really can't help it but they have to figure out what causes this. There may be a trivial answer, or something profound may be found out, who is to know in advance what will come of it. To everybody else the circumference of a circle may as well be three times its diameter.
- Eliezer 18y agoThe meaningfulness of rearranging the digits in a base-10 representation is fully and sufficiently demonstrated... by coming up with an interesting non-obvious mathematical fact that depends on rearranging the digits in a base-10 representation. Meaning is something you get out of a math trick, not something you put into it.