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Mysterious number 6174 (maths.org)
124 points by imgabe on Feb 13, 2009 | hide | past | web | favorite | 41 comments

It'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.

Define a function kaprekar(n, b) that will computer Kaprekar's operation on an integer n expressed in base b. Now we can explore the properties of this function. We know that if base = 10, the integer's digits in base 10 aren't all the same, and the integer has four digits, the function has a fixed point at 6174. Under what other bases and digit-counts does the function have a fixed point? Is it possible for kaprekar(a, b) to return an integer c with fewer digits than a, assuming once again that the digits are non-uniform? Etc.

"Number of digits in n" is shorthand for "floor(log(b)n) + 1" and the digits themselves can be expressed in terms of modulus (while n > 0: digits.append(n % b); n = n / b). Speaking in terms of digits just makes it easier to talk about what are really rather odd mathematical operations. It's true that their primary use is in our number representation system, but that doesn't make them any less interesting from a pure-mathematical viewpoint.

To 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.



What difference do his credentials make?

If you want to know if something is true, do research, don't just assume that someone's credentials make everything they say true.

Personal opinion coming up ...

I find this attitude an incredible strength in the hacker community, but also occasionally a real weakness. It leads to a complete dismissal of a source of information, and that, surely, can't be right - namely, the mindset and knowledge of the person making the statements. When making decisions you should use all the information available.

Yes, some information should be tagged as "hearsay" (you can't check the person's credentials) and some should be marked as "suspicious," but simply throwing it out is wrong.

When you teach or train it is critical that you know people's background so you can pitch the explanations and demonstrations at the right level. When you listen to someone, you should assess what level of confidence to place in what they say.

Work since Shannon has shown that the most efficient communication of information (as opposed to data) is achieved when the sender has an effective model of the receiver, and the receiver has an effective model of the sender.

Certainly many advances in breaking ciphers are achieved by knowing more about the source. Cryptographers know not to throw away information, even when it is suspect.

You can dismiss this, or go and research it. Does it sound interesting? Should you ignore it? Modern information theory says you shouldn't. You don't know my credentials, so you have to make that decision in a void. If I told you my credentials, would your opinion remain completely unchanged? Modern information theory says it shouldn't.

Yes, be skeptical, but use what information you have.

All of it.

ah, but then there is google:


"User Profile for: riderofgiraffes UserID: 383529 Name: Email: Registered: 1/11/07 Occupation: Mathematician Location: UK Total Posts: 350"

So much for that vague reputation, unless of course you've been fibbing ;) and / or someone else liked your moniker as much as you did, but I somehow doubt there are multiple riders of giraffes.

I fully agree with the above by the way, it feels just right.

I always tell my kid to distrust all sources of information, including his dad (there is a pardox in there somewhere) and to gather his own facts if the issue warrants the effort.

But such a search is uninformative for the author in question. If I had used a pseudonym you would have had no such additional information.

My point remains. With no credentials, with no idea of the author's background, you must perform independent verification. With some idea of the author's background, you have more information, and can decide whether the information is likely to be trustworthy, or at least worthy of pursuit or verification.

I am concerned that hacker culture teaches "Trust no one, verify everything." I think it is thereby unnecessarily impoverished.

I typed up a similar response.


I 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.

I am a graduate mathematics student, and you are embarrasingly wrong.

First, the term kernel certainly does have meaning in this context. The kernel of a mapping is the set of arguments it sends to the identity element (zero here), so in this case the kernel is just {1111, 2222, 3333, ... , 9999}. Like you, I thought the author should have used the term "fixed point," but there are situations where his way would be better. Details if anyone asks.

Now, why could anyone possibly gain from studying permutations on the digits of integers represented in base ten?

EDIT: the short answer is "cryptography." I'm cutting this post in two. For the longer answer, see below.

Here's the long answer to the question: what could anyone learn from rearranging the digits of an integer?

First off, it's helpful to know that the integers (denoted {Z}) are the canonical example of what's called ring structure, which basically means + and * work normally, but you aren't guaranteed clean division. Since they're simple, mathematicians sometimes use {Z} as a stand-in for more persnickety structures in the same way they use n-dimensional Euclidean space as a proxy for thinking about infinite-dimensional vector spaces. Articles like this one usually get written when an algebraist or number theorist publishes an epic proof, and the only way to make it comprehensible to a lay audience is to project it down into toy examples in a simple space. That's kind of a "trust me" answer by itself, of course. So without further ado, the math:

There are two notable things about rings. You can make polynomials with + and * in them, and division generally leaves you with remainders. Remainders are okay, though, because they let you define new rings. {Z mod 12 Z}, for example, is shorthand for "integers mod 12," or more formally, "map the integers to a new space where all the multiples of 12 get sent to zero," so {1, 2, ..., 11 } act normal, 12 is actually zero, 13=12+1=0+1=1 and so on.

It turns out that this is all you need to represent numbers in place-value notation. A number N in base r has a representation of n digits:

N = (A_n)(r^n) + ... + (A_2)(r^2) + (A_1)r + A_0

which is just a polynomial whose coefficients are members of the ring {Z mod r*Z}. If you map that back to the integers, plug in the actual value of r, and crunch the numbers, you get your original value of N back.

So, "rearranging the digits of an integer" is actually "permuting the coefficients of polynomials with coefficients mod r."

Now, suppose you have an email message that's 12000 characters long. You can think of that message as the polynomial representation, modulo 256, of a really big number. Mapping that number into different modular spaces, or permuting the coefficients of its various representations so that you can get it back later ("encrypting the message") forms the basis of the field of cryptography, which in turn is closely related to information theory and the theory of ergodic processes, etc, etc, etc.

For more details, you might start with:



I agree that if we decide that 0 is the identity element then the concept of the kernel does have meaning. Well except that the article says the map isn't defined over {1111, 2222, 3333, ... , 9999} However 0 is just the additive identity and there's really no reason to select addition. We could just as easily select the multiplicative identity (1). So yes if you arbitrarily select 0 and then redefine function you can form a concept of the kernel (it's pretty boring as far as I can see).

However the article does specifically say that 6174 is the kernel. Now for that to be correct under your definition it would need to be true that: 7641 - 1467 = 0 Which we all know to be false. Your post seems to suggest that you think this use of kernel was a poor word choice instead of completely incorrect. Perhaps you could send me the details that your post promises.

The base of ten is the most natural one (if I remember well, we all have about 10 fingers), and it is the most important one.

Remember that math is only of use if it can be applied to real world problems. Other bases than 10 are only interesting for computing machines, but even there the result have to be human readable (again base 10), otherwise nobody would be able to understand.

I see the base 10 as the only really important base to us humans (machines may have different preferences...).

The base of ten is the most natural one

It probably just seems that way because we're used to it. At some point when designing a system of numbers, 10 was a convenient answer to the question of how many digits to use. This doesn't mean it has a neurological basis.

I searched the Web a little bit for research pointing to a neurological basis for decimal. I didn't find it, but I did find this article on the brain's handling of numbers: http://arstechnica.com/science/news/2007/01/6661.ars

I see the base 10 as the only really important base to us humans (machines may have different preferences...)

If communication is ever established with non-human intelligences, it will be important to move beyond speciesism. When aliens finally land, or AI awakens, we should be able to say "Welcome. All your base are okay with us."

If they use a base based (no pun intended) number system at all.

Good point. I guess roman numerals would be an example of an un-based system, although it still seems to make you think in powers of ten.

See you again in 60 minutes..

What 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.


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.


Call 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?

A 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.

Okay 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.

Oh, 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").

A 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:


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."

6174 it's the only real solution of this system of equations: 9 ≥ a ≥ b ≥ c ≥ d ≥ 0 abcd - dcba = ABCD

Are you positive that's true for all bases?

The 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.

He 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]

No, 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.

I think we are talking past each other. In base ten if you start with 1234 and cycle though asdf - fdsa > new number you hit a single stable number 6174 .

In base 11 using 6174 you cycle:

4808, 87A3, 7094, 9272, 3098, 9452, 7094 (cycle)

I was thinking of a http://en.wikipedia.org/wiki/Logistic_map for this equation abcd - dcba > might be interesting across different bases.

Yes they are. Where he meant 1000a+100b+10c+d he wrote abcd.

Who 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?

Did you mean 3x^2?

I 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?

The numbers in the article aren't rearranged arbitrarily, they're sorted to ascending and descending order.

I 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?

Just 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.

The 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.

Downvoted for implying there's anything mathematically special or interesting about base 10. Okay, I can't really downvote, but you get the idea.

first 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.

"...it also has one more surprise up it's sleeve..."


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