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?
What real mathematical operation involves rearranging digits in an integer?
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.
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.
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").
for an example see:
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."
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?
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?
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?
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]
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.
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.