This 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)
Yes 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.
Say the correct value is 42. He broke the number down into 10x4 + 2. It is just writing the correct number in a form that emphasises the important elements for the transposition.
I 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.
I 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.
In a similar vein I got really hooked in school on Lychrel numbers => take a number x and reverse its digits to form y. Add x and y, repeat. Eventually this process leads to numbers that are palindromic (they are the same if you reverse their digits). Except some numbers like 196 do not seem to ever form a palindrome. No one knows if this is true or if the palindrome is so big that computers have yet to find it.
That'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.
One 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.
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.
Reminds 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.
"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)
Yeah, I asked ChatGPT 3.5 as well. My prompt though yielded: "Yes, what you're describing sounds like an exhibit known as the "Ratchet Effect" at the Exploratorium in the Bay Area."
6174 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, there is not a simple Kaprekar Constant in base 16.
Readit News
To prove why this is so:
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.
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.
Dead Comment
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.
Dead Comment
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.
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.
https://plus.maths.org/content/mysterious-number-6174
It does appear there are cycles for other lengths.
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.
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?