6174=Kaprekar's constant

flip2004

The number 6174 arises in the following famous problem :

Take any 4-digit number $x_1$ which uses more than one digit and find the difference $x_2:=T(x_1)$ between the numbers formed by writing the digits in descending order and ascending order.
For example, starting with $x_1= 1470$ yields $x_2=T(x_1)=7410- 0147=7263$ . Iterate this process using the difference x_2 as the new 4-digit number. In other words,
$\begin{array}{lcl} x_3&=&T(x_2)=7632-2367=5265 \\ x_4&=&T(x_3)=6552-2556=3996\\ x_5&=&T(x_4)=9963-369...$
The Indian mathematician D.R. Kaprekar discovered that this process leads in at most 7 steps to the number ${\mathbf 6174}=$ Kaprekar's constant, a fixed point of the iteration.
QUESTION: Justify the Kaprekar's routine. Try to find generalizations of this algorithm .

Remarks:
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.

**Posted:** Wed Mar 09, 2005 8:19 pm

spoudyal · Poincare Conjecture Offline Joined: 29 Dec 2004 Posts: 145

This is a bit out of my league. I'm sure that someone here could do this but I am the first person to reply since you posted this question last summer. Oh well, maybe everyone will look at it now?

**Posted:** Fri Apr 08, 2005 11:50 pm

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Jeremy · P versus NP Offline Joined: 25 Dec 2004 Posts: 24

holy crap

needless to say i dont know how but that is super neat

**Posted:** Tue Jun 14, 2005 4:20 pm

seamusoboyle

Well after the first iteration it has to be divisble by 9 anyway!

**Posted:** Sat Jul 09, 2005 1:39 pm

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