## 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 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.

In general, if the number is 1000a + 100b + 10c + d in which, a > b > c > d, ∀ 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.
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).
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 10s place and 4 at 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.

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.

0001, 1112 (& 1110), 2223 (& 2221), 3334 (& 3332), 4445 (& 4443), 5556 (& 5554), 6667 (& 6665), 7778 (& 7776), 8889 (& 8887). Here the number of iterations is 8.

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.

## Thursday, December 2, 2021

### 130-Basics of Trigonometry - 16

Basics of trigonometry

Now we will study some examples of trigonometry.

## Tuesday, March 9, 2021

### 127-EMI statement makers for all types of loan

PURPOSE/OBJECTIVES

This tool will help users to verify the EMI statement with the bank’s EMI statement. It will help users to prevent bank fraudulence if any.

I faced the same type of problem. A huge amount was included in my loan amount after the payments of 6 EMI's by one of the well-known finance company which could not be noticed by anyone. At the end of the year, my program was not tallying with the statements of the loan finance company. So, I re-checked my software, thinking that the bank may not do such types of mistakes. I modified my software and checked the statement. There was fraudulence in the bank's statement and the same was sorted out at that time and everything became normal as per my software. So, it will be very useful to everyone who takes a loan from the bank of any finance company.

Overview

For any loan taken, we must repay it with equated monthly installments called EMI. This tool requires limited information like:
1) The opening balance of a loan.
2) EMI-start-Month.
3) EMI amount.
4) Rate of interest.
5) Part payment amounts with month.
6) The details of EMI were skipped.

Based on your information, this tool will calculate the entire EMI account statement, which can be compared with the statement of the loan-providing company.

Procedure

### Step-1

Here, enter the principal amount (P), rate of interest (PCPA), (write 13 for 13%), and the number of months of EMI, then we will get EMI-amount. The information will be forwarded for the first-year EMI chart.

The software will take the principal-loan-amount, rate-of-interest, and EMI-start-month directly from the previous sheet. The latest modified information of these fields will be available throughout the EMI charts.

### Step-2

For any changes in the EMI amount, enter the month from the drop-down menu and the new EMI amount in the next cell under the head "EMI".

For any changes in the rate of interest, enter the month from the drop-down menu and the new rate of interest in the next cell under the head "INTEREST". If any part-payment is done, then enter the month from the drop-down menu and part-payment amount in this cell under head "Part-Payment". If any EMI is missed, it will be entered here under the head "EMI not paid for some month".

### Step-3

Summary of the EMI statement:

All the information is available with the graph.

ANIL SATPUTE

## Friday, November 27, 2020

### 126-Calculator of a commutation of a pension

Dear friends,

Now you can download FORM-1 from this site also. Go to the bottom, click on top-right-symbol which shows as "pop-out " and download the form.

This post is to calculate the commutation of a pension and pension amount after commutation. So many employees have taken VRS even at the age of 50 years. So, this software will help them to set a reminder for submission of Form-1 in their admin office. Following important particulars regarding the commutation of a pension will be available in this software.

#### Step-2:

This output will be the same as we get the details of the commutation of our pension. This software is with more information about the commutation of pension.

Thanks for using this software.

ANIL SATPUTE

## Tuesday, July 28, 2020

### 125-How to open your NPS Account Online

Dear friends,
Due to COVID-2019, it is always safe to stay at home. So this blog will help you to open your NPS account (National Pension System) online.

It will really help you a lot.

NPS Tier 1 Tax Benefits
You get a tax deduction under Section 80CCD(1) and 80CCD(2) on contributions to the NPS up to Rs 1.5 lakh per annum.
• In addition you can get a tax deduction on contributions up to Rs 50,000 under Section 80CCD(1B). Hence the total tax benefit on contributions to the NPS is Rs 2 lakh per annum.
• In addition, the returns on NPS are tax-exempt while the money is retained in the NPS account.
• On maturity, up to 60% of the accumulated NPS corpus can be withdrawn free of tax. Another 40% must be used to buy an annuity (regular pension).

When I was trying to open an on-line NPS account, I faced some problems and the same is being shared hare.

So we will see the procedure step by step:

#### Step-1:

Click on the following link to open the NPS account.

I used KARVY to register for NPS. You can choose anything.

#### Step-2:

Click on the Join NPS New User button.

#### Step-3:

If you select PAN, then the following task is being done.

NOTE: Please select the Point of Presence (POP) where you have an existing relationship – either a Savings/ Current account (in case of Bank) or any Demat/ Mutual Fund/ Insurance, etc. (in case of non-Bank). Your KYC verification under NPS will be done by the selected POP (Bank/ non-Bank). A onetime fee of up to maximum Rs. 125 (plus applicable taxes) will be charged by POP (Bank/ non-Bank) as KYC authentication charges.

Secondly, if you choose the AADHAR card, then follow the following diagram.

Here you need to be ready for the following:

#### Step-4:

Now generate an XML file of your AADHAR card which is to be uploaded while registering for the NPS account.

#### Step-5:

Now upload XML file on the system.

#### Step-6:

Now fill all the steps as given below.

#### Step-7:

Now you can pay a minimum of Rs. 500 and after payment, you will get your PRAN

#### Step-8:

Kindly note that your KYC details will be sent to the Bank selected by you during the registration process for verification. Once Bank has confirmed the KYC verification, you have the option to esign the registration form.

#### Step-9:

Now you can also download the NPS mobile app from Google Play Store for Android (http://bit.do/ewpU8) and Apple Play Store for iOS by searching "NPS by KFintech CRA".

In the same way, you can also register for the NPS account using the NSDL portal.

Thanks for visiting my site.