The Amazing Concepts

Happy number is a positive integer that gives the final number as 1 when certain steps ( square and sum the digits )   are performed continuously. Checking a Happy Number Step 1 : Consider any number. Step 2 : Square the number and replace the resultant with the sum of the squares of its digits. Step 3 : Repeat the steps. The final number will be 1, or the steps will loop endlessly in a cycle. Those numbers (considered in Step 1) that give the final number as 1 are Happy numbers . Examples Example 1 Step 1 : Consider number 1. Step 2 : Square the number => 1 2 => 1 x 1 => 1. Step 3 : Repeat the steps. 1 2 => 1 x 1 => 1 Repeating the steps gives the final number as 1. Therefore, the number considered in Step 1 is a Happy number . Example 2 Step 1 : Consider number 2. Step 2 : Square the number => 2 2 => 2 x 2 => 4. Step 3 : Repeat the steps. 4 2 => 4 x 4 => 16 1 2 + 6 2 => 1 + 36 => 37 3 2 + 7 2 => 9 + 49 => 58 5 2 + 8 => 2...

Dudeney number is a positive integer that is equal to the cube of the sum of all its digits. Checking a Dudeney Number Step 1 : Consider any number. Step 2 : Sum all its digits. Step 3 : Cube the number obtained in Step 2. If the number obtained in Step 3 is equal to the number considered in Step 1, it is a Dudeney number. But if the number obtained in Step 3 is not equal to the number considered in Step 1, it is not a Dudeney number. Examples Example 1 Step 1 : Consider number 0. Step 2 : Sum all its digits. The resultant is 0. Step 3 : Cube the number 0 obtained in Step 2 => 0 3 => 0 x 0 x 0. The resultant is 0. The number obtained in Step 3 is equal to the number considered in Step 1, hence 0 is a Dudeney number. Example 2 Step 1 : Consider number 1. Step 2 : Sum all its digits. The resultant is 1. Step 3 : Cube the number 1 obtained in Step 2 => 1 3 => 1 x 1 x 1. The resultant is 1. The number obtained in Step 3 is equal to the number considered in Step 1, hence 1 i...

Kaprekar number is one of the amazing concepts in Mathematics. The square of a number when divided into parts and none of the parts is equal to 0, and sum of both the parts forms the original number, then that number is a Kaprekar number . Let's understand this concept in detail: Checking a Kaprekar Number Following are the steps to check whether any number is a Kaprekar number or not: Step 1 : Consider any number N (must be greater than 0). Step 2 : Square the number N (N x N). Step 3 : Divide the resultant obtained in Step 2 into two parts (left and right). Possible scenarios that occur when dividing the resultant into two parts: Scenario 1 : If the resultant obtained in Step 2 is a single digit number If the resultant obtained in Step 2 is a single digit number (no matter even or odd), compare it with the number considered in Step 1.  If the number in Step 2 is equal to the number in Step 1, it is a Kaprekar number. But if the number  in Step 2 is not equal to the...

In Mathematics, one of the amazing concepts is the Reverse-Add-Then-Sort (RATS) Sequence. The RATS Sequence is a sequence that is formed by reversing , adding , then sorting the digits. Let's understand this concept in detail: Form the RATS Sequence Step 1 : Consider any number (must be greater than 0). Step 2 : Reverse the number. Step 3 : Add the original number (considered in Step 1) with the number formed by reversing the number (in Step 2). Step 4 : Sort the number obtained in Step 3 in ascending order. The resultant obtained after sorting the number forms the next sequence after n iteration. Examples First Iteration Step 1 : Number 1 . Step 2 : Reversing the number 1  gives the resultant  1 . Step 3 : Adding the original number 1 (considered in Step 1) with the number formed by reversing (in Step 2)  i.e. 1 gives the resultant 2 . Step 4 : Sorting the number obtained in Step 3 in ascending order gives the resultant 2 . Here number 2 forms the next sequence after...

You might have heard the term black hole . A black hole is a region of spacetime where gravity is so strong that even light or electromagnetic waves cannot escape it. In Mathematics, the number 123 is called the black hole number. Let’s understand this amazing concept in detail. The Procedure Step 1 : Consider any number . Step 2 : Count the number of even digits. Step 3 : Count the number of odd digits. Step 4 : Count the number of total digits. Step 5 : Consider the number (" even " " odd " " total ") obtained in the n   iteration. Step 6 : Repeat the steps ( Step 1 to Step 5 ) after every iteration until you get the number (" even " " odd " " total ") as  123 . Note : If you repeat the steps ( Step 1 to Step 5 ) even after getting the number 123, you will obtain the same number i.e. 123 after every iteration. Example First Iteration Step 1 : Number 1234567890 . Step 2 : Number of  even  digits = 5 . Step 3 : Number of...

A triangular number (triangle number) is a number that represents the pattern of dots arranged in a way that they form an equilateral triangle. An equilateral triangle (also known as a regular triangle) is the one in which all the three sides are of equal length. A triangular number is denoted by Tn that represents the number of dots. Mathematical Formula Below is the mathematical formula for a triangular number: where n is a positive number such as 1, 2, 3, 4, 5, ..... , n. Triangular Number Sequence Below is the triangular number sequence: [1, 3, 6, 10, 15, 21, 28, 36, ..... ] Example For n = 1 Putting this value of n in the above mathematical formula: T1 = 1(1 + 1)/2 T1 = 1(2)/2 T1 = 2/2 T1 = 1 For n = 2 Putting this value of n in the above mathematical formula: T2 = 2(2 + 1)/2 T2 = 2(3)/2 T2 = 6/2 T2 = 3 For n = 3 Putting this value of n in the above mathematical formula: T3 = 3(3 + 1)/2 T3 = 3(4)/2 T3 = 12/2 T3 = 6 For n = 4 Putting this value of n in the above mathematical formu...