Look at 6 and 9 digit numbers.. 6 and 9 are 2nd and 3rd multiples of 3..
Moreover, the results for 6 an 9 digits are all containing the numbers 495 or 6174 and then taking the 6 digit versions which add extra digits by putting the original numbers backwards interspaced with itself.. For 495, _4 _9_5 woven with 5_9_4_ becomes the mirrored number 549945…. and for 6174 then 6_17_4 and the added two digits 3 and 6 only _3__6_ which is only one off from backwards first and last digits of 4 and 6 becoming 631764.
Now for the related NINE digit versions, you take the six digit versions and stutter the numbers as such
549945 becomes 5_49_94_5 adding the backwards of 495 interspersed as _5__9__4_ to become 554999445 and
864197532 contains _641_753_. or 641753 which is 631764 only shifted up or down one for three of the digits.
Then the EIGHT and TEN digit numbers are related somehow to the 4 and 6 digits numbers for at least the first answer which is the same as the eight digit number answer just stuttering in another mirrored copy of the digits 3 and 6 again. The middle number unsure how that might derive till. The last ten digit answer clearly contains the last eight digit answer as such: 9975084201. =. _9750842_1 ….
And don't get me started on the patterns for the added digits and evens and odds… ;)
Your link to "Mathews Archive of Recreatrional Mathematics" is no longer good. The closest related page I could find by Googling was the one here:
I saw some mentions of Kaprekar snooping through lower links but none which addressed the numbers 495 or 6174.
OK, totally done trying to skull-out over this odd mathematical number and its relationships… ;)