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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:
https://www.primepuzzles.net/thepuzzlers/Schnider.htm
I saw some mentions of Kaprekar snooping through lower links but none which addressed the numbers 495 or 6174.
OK, totally done trying to skullout over this odd mathematical number and its relationships… ;)