Kaprekar constant

The 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 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 $b=10$ .

单词	KaprekarConstant
释义	Kaprekar constant The 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 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 $b=10$ .
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