## Miraculous Constant 8181

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

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 100’s placed digit will be 2.

**∀ a, b, c, d ∈**

**W, set of whole numbers,**then 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**

**Method**

**Figure-1**

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

**Standard patterns of numbers and their iterations**

**Set-1: One digit is repeated three times, and the fourth digit is different.**

1. **1000a + 100a + 10a + (a+1) for a = 0 and 1000a + 100a + 10a + (a±1) for 1 ≤ a ≤ 8 , ∀ a ∈ ****W, set of whole numbers****.**

*Note: In the 1st iteration, the Zigzag Number is smaller
than the Reverse Number, so a smaller number is subtracted from the bigger
number. So here the modulus value of -90 is taken. So, Mod (-90) = 90.*

2. **1000a
+ 100a + 10a + (a+2) for a = 0 or a = 1 and 1000a + 100a + 10a + (a±2) for 2 ≤
a ≤ 7, ∀ a ∈ ****W,
set of whole numbers****.**

0002, 1113, 2224 (& 2220), 3335
(& 3331), 4446 (& 4442), 5557 (& 5553), 6668 (& 6664), 7779
(& 7775). Here the number of iterations is 7.

3. **1000a
+ 100a + 10a + (a+3) for 0 ≤ a ≤ 2 and 1000a + 100a + 10a + (a±3) for 3 ≤ a ≤
6, ∀ a ∈ ****W, set of whole numbers****.**

0003, 1114, 2225, 3336 (& 3330),
4447 (& 4441), 5558 (& 5552), 6669 (& 6663). Here the number of
iterations is 5.

4. **1000a
+ 100a + 10a + (a+4) for 0 ≤ a ≤ 3 and 1000a + 100a + 10a + (a±4) for 4 ≤ a ≤
5, ∀ a ∈ ****W, set of whole numbers****.**

0004, 1115, 2226, 3337, 4448 (&
4440), 5559 (& 5551). Here the number of iterations is 6.

5. **1000a
+ 100a + 10a + (a+5) for 0 ≤ a ≤ 4, ∀ a ∈ ****W, set of whole numbers****.**

0005, 1116, 2227, 3338, 4449. Here
the number of iterations is 8.

6. **1000a
+ 100a + 10a + (a+6) for 0 ≤ a ≤ 3, ∀ a ∈ ****W, set of whole numbers****.**

0006, 1117, 2228, 3339. Here the
number of iterations is 8.

7. **1000a
+ 100a + 10a + (a+7) for 0 ≤ a ≤ 2, ∀ a ∈ ****W, set of whole numbers****.**

0007, 1118, 2229. Here the number of
iterations is 6.

8. **1000a
+ 100a + 10a + (a+8) for 0 ≤ a ≤ 1, ∀ a ∈ ****W, set of whole numbers****.**

0008, 1119. Here the number of
iterations is 5.

9. **1000a
+ 100a + 10a + (a+9) for a=0, ∀ a ∈ ****W, set of whole numbers****.**

0009. Here the number of iterations
is 7.

**Set-2: Two digits are repeated 2 times. Each number in every
group is incremented by 1111. So, each group has the same number of iterations.
**

1. **1000a
+ 100a + 10(a+1) + (a+1) for 0 ≤ a ≤ 8, ∀ a ∈ ****W, set of whole numbers****.**

0011, 1122, 2233, 3344, 4455, 5566,
6677, 7788, 8899. Here, the number of iterations is 2.

2. **1000a
+ 100a + 10(a+2) + (a+2) for 0 ≤ a ≤ 7, ∀ a ∈ ****W, set of whole numbers****.**

0022, 1133, 2244, 3355, 4466, 5577,
6688, 7799. Here the number of iterations is 6.

3. **1000a
+ 100a + 10(a+3) + (a+3) for 0 ≤ a ≤ 6, ∀ a ∈ ****W, set of whole numbers****.**

0033, 1144, 2255, 3366, 4477, 5588,
6699. Here the number of iterations is 4.

4. **1000a
+ 100a + 10(a+4) + (a+4) for 0 ≤ a ≤ 5, ∀ a ∈ ****W, set of whole numbers****.**

0044, 1155, 2266, 3377, 4488, 5599.
Here the number of iterations is 5.

5. **1000a
+ 100a + 10(a+5) + (a+5) for 0 ≤ a ≤ 4, ∀ a ∈ ****W, set of whole numbers****.**

0055, 1166, 2277, 3388, 4499. Here,
the number of iterations is 3.

6. **1000a
+ 100a + 10(a+6) + (a+6) for 0 ≤ a ≤ 3, ∀ a ∈ ****W, set of whole numbers****.**

0066, 1177, 2288, 3399. Here the
number of iterations is 3.

7. **1000a
+ 100a + 10(a+7) + (a+7) for 0 ≤ a ≤ 2, ∀ a ∈ ****W, set of whole numbers****.**

0077, 1188, 2299. Here the number of
iterations is 5.

8. **1000a
+ 100a + 10(a+8) + (a+8) for 0 ≤ a ≤ 1, ∀ a ∈ ****W, set of whole numbers****.**

0088, 1199. Here the number of
iterations is 4.

9. **1000a
+ 100a + 10(a+9) + (a+9) for a = 0, ∀ a ∈ ****W, set of whole numbers****.**

0099. Here the number of iterations
is 6.

**Set-3: One digit is repeated two times. See the following
form of numbers: Each number in every group is incremented by 1111. So, each
group has the same number of iterations.
**

1. **1000a
+ 100a + 10(a+1) + (a+2), 0 ≤ a ≤ 7, ∀ a ∈ ****W, set of whole numbers****.**

0012, 1123, 2234, 3345, 4456, 5567,
6678, 7789. Here the number of iterations is 5.

2. **1000a
+ 100a + 10(a+1) + (a+3), 0 ≤ a ≤ 6, ∀ a ∈ ****W, set of whole numbers****.**

0013, 1124, 2235, 3346, 4457, 5568,
6679. Here the number of iterations is 6.

3. **1000a
+ 100a + 10(a+1) + (a+4), 0 ≤ a ≤ 5, ∀ a ∈ ****W, set of whole numbers****.**

0014, 1125, 2236, 3347, 4458, 5569.
Here the number of iterations is 4.

4. **1000a
+ 100a + 10(a+1) + (a+5), 0 ≤ a ≤ 4, ∀ a ∈ ****W, set of whole numbers****.**

0015, 1126, 2237, 3348, 4459. Here
the number of iterations is 4.

5. **1000a
+ 100a + 10(a+1) + (a+6), 0 ≤ a ≤ 3, ∀ a ∈ ****W, set of whole numbers****.**

0016, 1127, 2238, 3349. Here the
number of iterations is 4.

6. **1000a
+ 100a + 10(a+1) + (a+7), 0 ≤ a ≤ 2, ∀ a ∈ ****W, set of whole numbers****.**

0017, 1128, 2239. Here the number of
iterations is 4.

7. **1000a
+ 100a + 10(a+1) + (a+8), 0 ≤ a ≤ 1, ∀ a ∈ ****W, set of whole numbers****.**

0018, 1129. Here the number of iterations
is 6.

8. **1000a
+ 100a + 10(a+1) + (a+9), a = 0, ∀ a ∈ ****W, set of whole numbers****.**

0019. Here the number of iterations
is 5.

**Reference**

1) The concept of Kaprekar constant 6174 - Numberphile.

**Keywords**

Numbers,
Fun with Mathematics, iterations, Zigzag Number, Reverse Number.

**Acknowledgments**

Thanks to: Jyoti Satpute, Aaswad Satpute, Priyanka Satpute, Abhishek Satpute.