Mathemagics 3: The unsolved mystery of 6174

6174 is a very interesting number. Well, what's so interesting about this number? Let's find out.

Here is another mysterious number for you. As its name suggests this number was discovered by an Indian mathematician Dattatreya Ramachndra Kaprekar. Before coming to the Kaprekar constant let us first know about Kaprekar number.

Kaprekar number

Mathematically, Kaprekar number is defined as a number K, such that

K = p + q

And K = p*10ⁿ + q

Didn’t get it? Don’t worry. Let’s understand it in language:

Consider a number K. Now square the number. If the square of the number contains 2n (even number of) digits then add the left n-digit number to the right n-digit number. And if the square is of 2n+1 (odd number of) digits, then add the right n-digit number to the left n+1 digit number. Now if the sum is K, it is called a Kaprekar number. Remember, if you get free zero, or 0 at the starting or ending, you must avoid it. You can see the example of 4879 below. 

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For example

9² = 81;                8 + 1 =9

297² = 88209;     88+209 = 297

4879² = 23804641;    238 + 4641 = 4879

First few Kaprekar numbers are 1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4879, 5050….

Kaprekar constant

6174 is called the Kaprekar constant. This constant was discovered by DR Kaprekar in 1949.

What’s so special about this number? Well, let’s see.

Take any four digit number.

Arrange them in ascending order and descending order.

Find the difference between the so arranged numbers.

Repeat this process multiple times. And you will end up getting 6174.

 I can assure you there would be no more than 7 steps.

For example

Taking a random number 2315:  Ascending order – 5321, descending order – 1235

Step 1: 5321 – 1235 = 4086

Step 2: 8640 – 0468 = 8172

Step 3: 8721 – 1278 = 7443

Step 4: 7443 – 3447 = 3996

Step 5: 9963 – 3699 = 6264

Step 6: 6642 – 2466 = 4176

Step 7: 7641 – 1467 = 6174

Kaprekar constant is not applicable on numbers with all same digits like 1111, 2222, 3333 and so on.

This was for four digit numbers. There are Kaprekar number for any other digit-numbers as well.

Here is a list:

Digits	Kernel
2	9
3	495
4	6174
5	None
6	549945, 631764
7	None
8	63317664, 97508421
9	554999445, 864197532
10	6333176664, 9753086421, 9975084201

Take an example:

2-digit number:

91 – 19 = 72

72 – 27 = 45

54 – 45 = 9

I assure you this will not take more than 3 steps

3-digit number:

543 – 345 = 198

981 – 189 = 792

972 – 279 = 693

963 – 369 = 594

954 – 459 = 495

I assure you this will not take more than 5 steps.

What's the mystery behind this number? 

Well, the mystery is that the mystery has not been solved yet. Yes, it seems so lucrative and beautiful for the mathematicians - even for me (though I'm not one) - but it has remained unsolved till the date. So here is a chance for you to show the world your mathematical ability. Well, it's quite possible that if this mystery is solved a big theorem in number theory is just waiting for us.

So that’s all from the Kaprekar constant.