Consider a $n$-digit number ($N$), construct the biggest ($B$) and the smallest number ($S$) in decimal representation using these digits of $N$. Get $N_1=B-S$, repeat this process get $B_1,S_1,N_2$, so on and so forth.
According to the Indian Mathematician D.R. Kaprekar for the case of $n=4$ (4 digit numbers not all same) the process ($N_k$) always converges at the well known unique Kaprekar-constant: $6174$, after some $k$ iterations.
For 3-digit numbers, Kaprekar-constant is found to be 495.
I along with a 6th grader (Arjun Pandey) find that for 2-digit numbers the process oscillates and ends at one of the minimal 2-digit numbers: $18,27,36,45$; whereas $54,63,72,81$ are also meant but they are non-minimal.
Next, we find that in case of 5-digit numbers The process oscillates to end at one of the minimal 5-digit numbers: 13799, 14679, 22689, 23589, 24669, 33579, 34479, 34569, 35555, 45999. Any other permutation of digits in these numbers is also meant and we call them non-minimal, in all then there will be some 685$~~$ 5-digit Kaprekar-constants.
These Kaprekar-constants for 5-digit numbers are known in various permutations:
https://www.google.com/search?client=firefox-b-d&q=5+digit+kaprekar+number
We feel that the concept of minimal Kaprekar-constants in case of 2- and 5- digit numbers is more convenient and helpful.
Any comment in this regard is welcome.
