## Miraculous Constant 8181

**This work is a special birthday gift to my sweetheart, my wife Jyoti Satpute**

The ‘Kaprekar
constant’ (6174) is a constant with special properties. The ‘Miraculous
Constant’ is based on the ‘Kaprekar Constant’ and exhibits some special
properties in addition to the properties of the ‘Kaprekar Constant’.

For any four-digit
number that is made up of at least one differing digit, rearrange the digits as
follows:

1) Place the largest digit at the 1000’s place and the smallest digit at the 100’s place.

Out of the remaining 2 digits:

2) Place the largest digit at the 10’s place and the smallest digit at the unit place.

Let's refer to this
rearranged number as a Zigzag number.

Example: Let’s say the input number is 8492. Let’s form the Zigzag number from this.

Solution:

1) The highest digit of the given number 8492 is 9 and the smallest digit is 2. So, the 1000’s placed digit of the Zigzag number will be 9, and the 100’s placed digit will be 2.

2)
Now we
need to place the digits 8 and 4 where 8 is the greater and 4 is the smaller
digit, so 10’s placed digit will be 8 and the units digit will be 4.

3)
So, here
the zigzag number of 8492 will be 9284.

4)
Its
reverse number will be 4829.

1) Place the largest digit at the 1000’s place and the smallest digit at the 100’s place.

Out of the remaining 2 digits:

2) Place the largest digit at the 10’s place and the smallest digit at the unit place.

Example: Let’s say the input number is 8492. Let’s form the Zigzag number from this.

1) The highest digit of the given number 8492 is 9 and the smallest digit is 2. So, the 1000’s placed digit of the Zigzag number will be 9, and the 100’s placed digit will be 2.

In general, if the number is 1000a + 100b +
10c + d in which, a > b > c > d,

Now let us take the
positive difference between the “Zigzag number” and the “reverse number”.

Repeat this process called iteration until you get 8181 (Miraculous constant).

**∀ a, b, c, d ∈****W, a set of whole numbers,**then the Zigzag number will be 1000a + 100d + 10b + c and its reverse number is 1000c + 100b + 10d + a. At least one digit out of a, b, c, or d must be different from others.Repeat this process called iteration until you get 8181 (Miraculous constant).

**Introduction**

Kaprekar's constant is 6174. Our
interest is to develop some other criteria to get the Miraculous Constant 8181.
Let us take any 4-digit number with at least one different digit. We can take
any number like 0001 which has 4 digits and one digit is different.

**Method**

Let us take any 4-digit number,
8492. Here, the largest digit is 9. So, the thousand-place digit will be 9. The
smallest digit is 2, so the 100s place digit is 2. From 8492, digits 9 and 2
are being used, so the remaining digits are 8 and 4. To place a larger digit, 8
at the 10s place and 4 at the units place. So, our Zigzag number is 9284 and its
reverse number is 4829 and we get a positive difference.

**Figure-1**

*The number of iterations for 8492 to get the Miraculous Constant 8181 is 8.*

Here in iteration 3, the number 8181
is obtained. Again, from 8181 to reach 8181 we have 5 iterations, and they are
fixed.