6174
en.wikipedia.org
en.wikipedia.org
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)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)
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.
It’s not obvious to me at all, I had to think pretty hard about it.
For example if you accidentally swapped 210,00 to 120,00: 20x10 + 10 is the correct number, 20 + 10x10 is the swapped one.
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.
That is to say, ambition, distraction, uglification, and derision.
Mysterious number 6174 - https://news.ycombinator.com/item?id=2625832 - June 2011 (64 comments)
6174 - https://news.ycombinator.com/item?id=1625606 - Aug 2010 (1 comment)
Mysterious number 6174 - https://news.ycombinator.com/item?id=480200 - Feb 2009 (41 comments)
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.
It does appear there are cycles for other lengths.
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...
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).
> 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
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.
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.
[1] https://philosophy.unc.edu/wp-content/uploads/sites/122/2013...
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.
I think we're both being BS'ed.
Maybe archive.org would have something.
Possibly hallucinaited (sic).
Tangentially, how much other research gets lost in the ether because it wasn't as interesting as this.
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?
74943 -> 62964 -> 71973 -> 83952 -> repeat
63954 -> 61974 -> 82962 -> 75933 -> repeat
53955 -> 59994 -> repeat
That said, https://en.wikipedia.org/wiki/List_of_numbers is woefully incomplete.
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.
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.
("Binary digit" and "hexadecimal digit" are weird terms that abuse the language a bit.)
I see what you did here
Lots of people seem to think that, hence the -2 rating of my comment, but that's not the modern definition: https://en.wikipedia.org/wiki/Numerical_digit
Similar numbers (I presume) exist for other number bases, and it's an interesting question of whether they constitute some sort of strange attractor. istm quite a few mathematical discoveries have emerged from just farting around with inconsequential-seeming numerical oddities.
I do feel your frustration though. I'm into electronic music and math, but I regularly run into people who insist that tuning to 432hz instead of 440hz (the common default for western tonality) is better because 432 is numerologically interesting. I've wasted a lot of time trying to persuade people that yes, 432 is a very cool number, but the interval of a second (from which we derive tuning frequencies) is fundamentally arbitrary. I suppose it's true that if you tune everything slightly flat people will subconsciously feel like time is expanding, man.
You might be able to satisfy yourself by replacing "the digits of" with "the decimal digits in the base-10 representation of".
Happened across a neat comment yesterday that presents a defense of ten. Not 100% convinced but it is interesting to see pushback.
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?
And you can swear by that, if you know what I mean.
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* * * *The point of this trickery is that N-1 added to any number is really adding N (which adds 1 to the second position, by definition) and adding -1 (which subtracts 1 from the first position).
In base 10, this is the adding 9 trick. It can be extended by using any multiple of 9. That applies to the N-1 version, so that adding M*(N-1) to a base N number yields the same digit sum.
1+9 = 10 = 1
1 + 27 = 28 = 10 = 1
In hex:
1 + F = 10 = 1
1 + 2D = 2E = 10 = 1
1 sums digitwise to 1
1 + (10-1) = 10 which also sums to 1 in the same way
For example, per another's link in these comments, this 'trick' works for 3 digits, but hits 1 of 3 possible loops for 5 digits. From this, interesting but likely useless questions can arise, such as finding an easy way to test for these loops, seeing if there is a way to calculate the loop without brute forcing it, and understanding the problem enough to know how much of this holds true when swapping to a new base.
In general, most of this is just for fun and doesn't lead to anything serious. But sometimes a fun problem can be hard to solve, possibly leading to discovering something new, which ends up being applicable to more serious mathematics. Other times it can become a trap that just seems to waste time without ever leading anywhere, like the 3n+1 problem.
I don't think this should be considered numerology, though I do think sometimes people treat tricks as if they have some more serious meaning that they don't deserve, at least not based on how they are presented. 3 Blue 1 Brown goes into the spiral pattern of the primes as something that appears to be deep, but ends up being an unique way to present an otherwise boring tidbit about prime numbers.
Being further pedantic - aren't all digits base ten? I thought that was part of the definition of digit.
Other bases would have different words for their numbers - bit in binary, for example (which, yeah, I know, it a combination of the words "binary" and "digit").
Do we have another example? I don't think there are special terms for "octal digits" or "hexadecimal digits".
We call computer circuits "digital" even though they work in base 2.
Regardless of the word's origin, digits are simply the symbols in a positional number system: https://en.wikipedia.org/wiki/Numerical_digit
An argument against being overly pedantic in this case is that this is a neat and accessible example of something quirky about numbers, and so even people who don't know much about numbering systems can approach it. If you instead emphasize that it's base 10 or that there is "no such thing as a 4 digit number", the main thing you'll probably do is cause disinterest in anyone who is sometimes overwhelmed by math. :)
Randomly, one of my sons told me about 6174 just a week ago, and it turned into an interesting conversation following by a little programming to find more of these numbers. After we went down that rabbit hole for awhile, then the conversation shifted to how these numbers might look in e.g. hexadecimal, and that seemed about the right time for that topic to come up.
The point of the parent comment is that this is not a property of numbers in general. It's just a coincidence that only works in base-10.
For example, a prime number is prime in every base. An irrational number is irrational in every base. Collatz conjecture is valid in every base. This one is not.
What? Not at all. In fact, trying it in other bases, as well as with other numbers of digits (in both base 10 and other bases), is a useful way to get some insights into why it happens.
So, yes, the described "special" thing about 6174 is actually a special thing about the string 6174 (representing a number in base-10). And I'd say the fact so many people in this very thread don't understand it is exactly the proof that the GPs comment actually has some merit. People kinda mix up properties of numbers and properties of some other mathematical objects — like their representations in base-10. Most of numerological games are concerned with the latter. Which is why it's especially interesting, when something like that happens to hold in other bases, which sadly just isn't the case with Kaprekar's constant.
Finally someone noticed the irony, thank you.
(On the other hand, at least some people also criticize my tone rather than the point I'm making, which I guess is fair as well.)
Here are some additional examples:
dec, 3 digits: 495
hex, 3 digits: 7F8
hex, 30 digits: EECCAA88664421FFDDBB9977553312
dec, 30 digits: 988766544332209987766554332111
hex, 100 digits: FFFFEEEEDDCCCCBBAAAA9998888776666554444333222210FFFFEEDDDDCCCBBBBAA999988777766655554433332211110001
dec, 100 digits: 9999998888888877666666665544444444332222222210999999998877777777665555555544333333332211111111000001
Comparing this to numerology is just combative and doesn't help get your point across (as you can see by the downvotes).
Besides, going to a place where people are discussing something fun and explaining to them why it isn't really fun is just not a good way to get points across to people, no matter how valid you think they are.
A much better way to approach this IMO - don't say this is wrong, give something analogous that would work for all bases, which by the way would teach people this concept. E.g. extending "do all digits appear infinitely and evenly in the decimal representation of pi" to talking about "normal numbers".
But it's not like it's somehow less worthy than other mathematical games. After all, there could have been some meaningful property hidden in there. Doesn't appear so in this case, but you'd never know beforehand.
Numerology is far stupider than this admittedly useless arithmetic game.
no that's highly opinionated compressionn in the domain of crazy