Kaprekar's routine

[Tomek]

Is there an insightful proof of convergence
of Kaprekar's routine for four digit numbers ?

Regards,
Tomek

[alexand...@ulbsibiu.ro]

6174=Kaprekar constant
===================
1) Regarding Kaprekar's original publications:
He self-published most of his results, via a small
Indian publishing
company at his own expense.
2) It seems that by using Kaprekar's routine , exactly 77 four-digit
numbers,
namely 1000, 1011, 1101, 1110, 1111, 1112, 1121, 1211, ...
(see [13] Sloane's A069746), reach 0, while the remainder give

6174 in at most 8 iterations.

REFERENCES:
[1] Deutsch D. and Goldman B. , Kaprekar's Constant,
Math. Teacher 98,(2004) 234--242.
[2] Eldridge, K. E. and Sagong, S. The Determination of Kaprekar
Convergence
and Loop Convergence of All 3-Digit Numbers,
Amer. Math. Monthly 95, (1988) 105--112.
[3] Furno A.L. , J. Number Theory 13, no.2, (1981) 255--261.
[4] Hasse H. , Iterierter Differenzbetrag für $2$-stellige
$g$-adische Zahlen,
(German, Spanish summary)
Rev. Real Acad. Cienc. Exact. Fís. Natur. Madrid 72 (1978),
no. 2, 221--240.
[5] Hasse H. and Prichett G. D. ,
The determination of all four-digit Kaprekar constants,
J. Reine Angew. Math. 299/300 (1978), 113--124.
[6] Kaprekar D. R. , An Interesting Property of the Number 6174,
Scripta Math. 15,(1955) 244--245.
[7] Kiyoshi Iseki , Note on Kaprekar's constant,
Math. Japon. 29 (1984), no. 2, 237--239.
[8] Lapenta J. F., Ludington A. L. and Prichett G. D.,
An algorithm to determine self-producing r -digit g-adic
integers,
J. Reine Angew. Math. 310 (1979) 100--110.
[9] Ludington Anne L., A bound on Kaprekar constants,
J. Reine Angew. Math. 310 (1979), 196--203.
[10] Prichett G. D., Ludington A. L. and Lapenta J. F.,
The determination of all decadic Kaprekar constants,
Fibonacci Quart. 19 , no. 1, ,(1981) 45--52.
[11] Rosen Ken, Elementary Number Theory Book ,
(4-th. ed, see page 47)
[12] A. Rosenfeld, Scripta Mathematica 15 (1949), 241--246,
[13] Sloane, N. J. A. Sequences A069746,A090429, A099009,
and A099010 ,
in The On-Line Encyclopedia of Integer Sequences,
http://www.research.att.com/~njas/sequences/.
[14] Trigg, C. W. , All Three-Digit Integers Lead to...,
The Math. Teacher, 67 (1974) 41-45.
[15] Young, A. L. , A Variation on the 2-digit Kaprekar Routine,
Fibonacci Quart. 31,(1993) 138--145.

[drmw...@gmail.com]

May I pick a nit?
I do not believe the statement about 77 seeds leading to 0 can be
correct. The process is:

Take four-digit number and regard as though a string.
Write largest permutation and smallest permutation (padding with zeros,
as needed).
Subtract (largest minus smallest).
Iterate.

It should be clear that one can obtain zero if and only if one has
largest permutation = smallest permutation, making all four digits the
same. In fact, I believe that one of Ross Honsberger's Mathematics Gems
book specifies eliminating only the four-digit strings using just one
digit, with 6174 obtained within 7 interations.

Incidentally, by making this into a dynamic process, one can regard
Kaprekar's example as what I call a Mathemagical Black Hole. For info,
see http://members.aol.com/DrMWEcker/mathhole.html
(or select Mathemagical Black Hole from my main site,
http://beam.to/REC).

Best, Mike

Dr. Michael W. Ecker, Editor
Recreational & Educational Computing
DrMWEcker at aol.com