Mysterious Number 6174 - Kaprekar Number 6174 Theory

Mysterious Number 6174

The number 6174 is a really mysterious number. At first glance, it might not seem so obvious. But as we are about to see, anyone who can subtract can uncover the mystery that makes 6174 so special.

Kaprekar's operation

In 1949 the mathematician D. R. Kaprekar  from Devlali, India, devised a process now known as Kaprekar's operation. 

Rules


   Step 1  
 Select any four digit number, but do not select the ones in which all the four digits are the same like 1111, 2222, ...  


   Step 2  
 Arrange the digits in decreasing order. 


   Step 3  
 Arrange the digits in increasing order. 


   Step 4  
 Subtract the smaller number from the larger number. 


   Step 5  
 Repeat the steps 2, 3 and 4 until you get 6174 in repetition. 


Example
  
  Step : 1 ==> 4 digit number is 5620. 
  Step : 2 ==> Arrange decreasing order : 6520 
  Step : 3 ==> Arrange increasing order : 0256 
  Step : 4 ==> Subtract the smaller number from the larger     
number . 
          6520 - 0256 = 6264 
  Step : 5 ==> Take the final result and repeat the steps 2, 3, and 4 until
   you get the repetition of 6174. 


  Like,
  
   6264 = 6642 - 2466 = 4176 
   4176 = 7641 - 1467 = 6174 
   6174 = 7641 - 1467 = 6174 


Have fun with this math trick and discover why all the 4 digit numbers on subtracting using the kaprekar operation reach to the mysterious number 6174.

A very mysterious number...

Few Example :
              When we started with 2005 the process reached 6174 in seven steps, and for 1789 in three steps. In fact, you reach 6174 for all four digit numbers that don't have all the digits the same. It's marvellous, isn't it? Kaprekar's operation is so simple but uncovers such an interesting result. And this will become even more intriguing when we think about the reason why all four digit numbers reach this mysterious number 6174.

Kaprekar Number

          Another class of numbers Kaprekar described are the Kaprekar numbers.A Kaprekar number is a positive integer with the property that if it is squared, then its representation can be partitioned into two positive integer parts whose sum is equal to the original number (e.g. 45, since 45^2=2025, and 20+25=45, also 9, 55, 99 etc.) However, note the restriction that the two numbers are positive non-zero numbers ; for example, 100 is not a Kaprekar number even though 100^2=10000, and 100+00 = 100. This operation, of taking the rightmost digits of a square, and adding it to the integer formed by the leftmost digits, is known as the Kaprekar operation.

Number	Square	Decomposition
703	703² = 494209	494+209 = 703
2728	2728² = 7441984	744+1984 = 2728
5292	5292² = 28005264	28+005264 = 5292
857143	857143² = 734694122449	734694+122449 = 857143