KaprekarConstant 2006-09-20 18:37:13 2006-09-20 18:42:13 Definition Kaprekar routine % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here The {\em Kaprekar constant} $K_k$ in a given base $b$ is a $k$-digit number $K$ such that subjecting any other $k$-digit number $n$ (except the repunit $R_k$ and numbers with $k - 1$ repeated digits) to the following process: 1. Arrange the digits of $n$ in ascending order, forming the $k$-digit number $a$, and then in descending order, forming the $k$-digit number $b$. 2. If $a > b$, calculate $a - b = c$; otherwise $b - a = c$. 3. Goto step 1 using $c$ instead of $n$. eventually gives $K$. (This process is sometimes called the {\em Kaprekar routine}). For $b = 10$, the Kaprekar constant for $k = 4$ is 6174. Using $n = 1729$, we find that 9721 - 1279 gives 8442. Then 8442 - 2448 = 5994. Then 9954 - 4599 gives 5355. Then 5553 - 3555 gives 1998. Then 9981 - 1899 gives 8082. Then 8820 - 288 gives 8532. Then 8532 - 2538 finally gives 6174. (Some numbers take longer than others). $K_2$ and $K_7$ don't exist for $b = 10$.