Showing posts with label (006) Miraculous World of Numbers. Show all posts
Showing posts with label (006) Miraculous World of Numbers. Show all posts

Saturday, December 10, 2022

136-Number pattern and its beauty

Nowadays, the following pattern is being observed on some social networks.
Nowadays, the next convention is being observed on some social networks.
Now we will see the basics behind this pattern of numbers.

For any positive integer "a" 
(100a+10a+a)/(a+a+a) = 111a/3a ---------- equation-1
                                     = 111/3               
                                     = 37. 
It's always 37 for such types of patterns of the problem.
Putting a = 1, in equation-1, we get 111/3 = 37.
Similarly, putting a = 2, we get 222/6 = 37, and so on.

These are simply the basics and everyone knows these basics of numbers. Now we will see some different patterns of numbers.

2 digits form:

If we take the same pattern for 2 digits, it will look like the following figure.
For any positive integer "a" 
(10a+a)/(a+a) = 11a/2a ---------- equation-2
                       = 11/2               
                       = 5.5. 
It's always 5.5 for such types of patterns of the problem.
Putting a = 1, in equation-2, we get 11/2 = 5.5.
Similarly, putting a = 2, we get 22/4 = 5.5, and so on.

4 digits form:

For any positive integer "a" 
(1000a+100a+10a+a)/(a+a+a+a) = 1111a/4a ---------- equation-3
                       = 1111/4               
                       = 277.75. 
It's always 277.75 for such types of patterns of the problem.
Putting a = 1, in equation-3, we get 1111/4 = 277.75.
Similarly, putting a = 2, we get 2222/8 = 277.75, and so on.

Note: The same explanation is there for all the following figures.

5 digits form:

6 digits form:

7 digits form:

8 digits form:

9 digits form:

The same concept can be applied to any number of digits. This is only the beauty of numbers.

There is nothing new. It is a well-known fact and the basics of numbers.

Sunday, November 27, 2022

135-Magic cube part-6

We have already seen the preparation of the magic cube.
Today we will discuss some specific addition patterns of this magic cube. See the following figure of the magic cube.


Here see all the boxes with all corresponding numbers. Now we study "Top-Bottom-Front-Left-Diagonal-strip". See the following diagram taken from the above magic cube.


Here the addition of all the corresponding numbers will be 1164.

Tuesday, November 8, 2022

132-Magic cube part-5

Click here for the previous blog on magic cube.

We have already seen the preparation of the magic cube.

Today we will discuss some specific addition patterns of this magic cube. See the following figure of the magic cube.


Here see all the boxes with all corresponding numbers. Now we study "Top-bottom-diagonal-front-right-strip". See the following diagram taken from the above magic cube.

Here the addition of all the corresponding numbers will be 1164.

Click here for the next blog on magic cube.

Sunday, May 1, 2022

131-Miraculous Constant 8181

 Miraculous Constant 8181


The number 6174, well-known as the Kaprekar Constant, possesses unique mathematical characteristics that have captivated mathematicians for many years. Drawing inspiration from this idea, the Miraculous Constant 8181 is an astonishing numerical occurrence. This constant adherence to a specific iterative method uncovers concealed patterns and fascinating mathematical connections.


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, with 8 being the greater and 4 being the smaller digit. Therefore, the 10’s place digit will be 8, and the units digit will be 4.
3) Therefore, 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, 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 the others.
Now let us take the positive difference between the “Zigzag number” and the “reverse number”.
Repeat this process, called iteration, until you obtain the miraculous constant 8181.
Introduction
Kaprekar's constant is 6174. Our interest is in developing additional criteria to obtain the Miraculous Constant 8181. Let us take any 4-digit number with at least one different digit. We can take any number, such as 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 10's place and 4 at the units place. So, our Zigzag number is 9284, and its reverse number is 4829, resulting in 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.

Wednesday, February 26, 2020

119 Divisibility test of 7

Dear friends,

When I was in 7th grade in 1972, my teacher taught me the divisibility tests of 2, 3, 4, 5, 6, 8, 9, 10,  except 7, then I asked her the divisibility test of 7. On this, she told that there is no test available for 7. I took it as a challenge.

While studying the table of 7 carefully, one day I fount the wonderful method for the divisibility test of 7. Actually, it's very interesting.

This method is published in my book titled "JADU NAGARI SANKHYANCHI". (Magic world of numbers) written in Marathi by "Bruhan Mumbai Ganit Adhyapak Mandal, Mumbai" published on 26th Dec 1980.

Procedure:

1) Take a number.
2) Write down its unit placed digit.
3) Double this unit placed digit.
4) Write down our new number excluding unit placed digit.
5) Subtract double of the unit placed digit from our new number.
6) Repeat this till we get a 2 or 1 digit number. If this number is divisible by 7, then our original number is also divisible by 7.

Example-1:


Check the divisibility of 7 for the number:
6902
1
Your Number
A
6902
6902
2
Its Unit Placed Digit
B
2
2
3
Double of Unit placed Digit
2B
2x2
4
4
Your new Number excluding Unit Placed Digit
C
690
690
5
Subtract double of unit placed digit from remaining number
C-2B
690-4
686
Repeat the same procedure
6
Your Number
A
686
686
7
Its Unit Placed Digit
B
6
6
8
Double of Unit placed Digit
2B
2x6
12
9
Your new Number excluding Unit Placed Digit
C
68
68
10
Subtract double of unit placed digit from remaining number
C-2B
68-12
56
As 56 is divisible by 7, our number 6902 is also divisible by 7.

Example-2:


Check the divisibility of 7 for the number:
1981
1
Your Number
A
1981
1981
2
Its Unit Placed Digit
B
1
1
3
Double of Unit placed Digit
2B
2x1
2
4
Your new Number excluding Unit Placed Digit
C
198
198
5
Subtract double of unit placed digit from remaining number
C-2B
198-2
196
Repeat the same procedure
6
Your Number
A
196
196
7
Its Unit Placed Digit
B
6
6
8
Double of Unit placed Digit
2B
2x6
12
9
Your new Number excluding Unit Placed Digit
C
19
19
10
Subtract double of unit placed digit from remaining number
C-2B
19-12
7
As 7 is divisible by 7, our number 1981 is also divisible by 7.

Example-3:


Check the divisibility of 7 for the number:
20237
1
Your Number
A
20237
20237
2
Its Unit Placed Digit
B
7
7
3
Double of Unit placed Digit
2B
2x7
14
4
Your new Number excluding Unit Placed Digit
C
2023
2023
5
Subtract double of unit placed digit from remaining number
C-2B
2023-14
2009
Repeat the same procedure
6
Your Number
A
2009
2009
7
Its Unit Placed Digit
B
9
9
8
Double of Unit placed Digit
2B
2x9
18
9
Your new Number excluding Unit Placed Digit
C
200
200
10
Subtract double of unit placed digit from remaining number
C-2B
200-18
182
Repeat the same procedure
11
Your Number
A
182
182
12
Its Unit Placed Digit
B
2
2
13
Double of Unit placed Digit
2B
2x2
4
14
Your new Number excluding Unit Placed Digit
C
18
18
15
Subtract double of unit placed digit from remaining number
C-2B
18-4
14
As 14 is divisible by 7, our number 20237 is also divisible by 7.

In the same way, we can check the divisibility test for any number.

ANIL SATPUTE