Skip to main content

Happy Number

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 => 12 => 1 x 1 => 1.

Step 3: Repeat the steps.

12 => 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 => 22 => 2 x 2 => 4.

Step 3: Repeat the steps.

42 => 4 x 4 => 16

12 + 62 => 1 + 36 => 37

32 + 72 => 9 + 49 => 58

52 + 8 => 25 + 64 => 89

82 + 92 => 64 + 81 => 145

12 + 42 + 52 => 1 + 16 + 25 => 42

42 + 22 => 16 + 4 => 20

22 + 02 => 4 + 0 => 4

42 => 16

12 + 62 => 1 + 36 => 37

Repeating the steps does not give the final number as 1 and loop endlessly in a cycle. Therefore, the number considered in Step 1 is not a Happy number.

Example 3

Step 1: Consider number 3.

Step 2: Square the number => 32 => 3 x 3 => 9.

Step 3: Repeat the steps.

92 => 9 x 9 => 81

82 + 12 => 64 + 1 => 65

62 + 52 => 36 + 25 => 61

62 + 12 => 36 + 1 => 37

32 + 72 => 9 + 49 => 58

52 + 82 => 25 + 64 => 89

82 + 92 => 64 + 81 => 145

12 + 42 + 52 => 1 + 16 + 25 => 42

42 + 22 => 16 + 4 => 20

22 + 02 => 4 + 0 => 4

42 => 16

12 + 62 => 1 + 36 => 37

32 + 72 => 9 + 49 => 58

Repeating the steps does not give the final number as 1 and loop endlessly in a cycle. Therefore, the number considered in Step 1 is not a Happy number.

Example 4

Step 1: Consider number 4.

Step 2: Square the number => 42 => 4 x 4 => 16.

Step 3: Repeat the steps.

162 => 16 x 16 => 256

22 + 52 + 62 => 4 + 25 + 36 => 65

62 + 52 => 36 + 25 => 61

62 + 12 => 36 + 1 => 37

32 + 72 => 9 + 49 => 58

52 + 82 => 25 + 64 => 89

82 + 92 => 64 + 81 => 145

12 + 42 + 52 => 1 + 16 + 25 => 42

42 + 22 => 16 + 4 => 20

22 + 02 => 4 + 0 => 4

42 => 16

162 => 16 x 16 => 256

22 + 52 + 62 => 4 + 25 + 36 => 65

Repeating the steps does not give the final number as 1 and loop endlessly in a cycle. Therefore, the number considered in Step 1 is not a Happy number.

Example 5

Step 1: Consider number 5.

Step 2: Square the number => 52 => 5 x 5 => 25.

Step 3: Repeat the steps.

252 => 25 x 25 => 625

62 + 22 + 52 => 36 + 4 + 25 => 65

62 + 52 => 36 + 25 => 61

62 + 12 => 36 + 1 => 37

32 + 72 => 9 + 49 => 58

52 + 82 => 25 + 64 => 89

82 + 92 => 64 + 81 => 145

12 + 42 + 52 => 1 + 16 + 25 => 42

42 + 22 => 16 + 4 => 20

22 + 02 => 4 + 0 => 4

42 => 16

12 + 62 => 1 + 36 => 37

32 + 72 => 9 + 49 => 58

Repeating the steps does not give the final number as 1 and loop endlessly in a cycle. Therefore, the number considered in Step 1 is not a Happy number.

Example 6

Step 1: Consider number 6.

Step 2: Square the number => 62 => 6 x 6 => 36.

Step 3: Repeat the steps.

362 => 36 x 36 => 1296

12 + 22 + 92 + 62 => 1 + 4 + 81 + 36 => 122

12 + 22 + 22 => 1 + 4 + 4 => 9

92 => 81

82 + 12 => 64 + 1 => 65

62 + 52 => 36 + 25 => 61

62 + 12 => 36 + 1 => 37

32 + 72 => 9 + 49 => 58

52 + 82 => 25 + 64 => 89

82 + 92 => 64 + 81 => 145

12 + 42 + 52 => 1 + 16 + 25 => 42

42 + 22 => 16 + 4 => 20

22 + 02 => 4 + 0 => 4

42 => 16

12 + 62 => 1 + 36 => 37

32 + 72 => 9 + 49 => 58

Repeating the steps does not give the final number as 1 and loop endlessly in a cycle. Therefore, the number considered in Step 1 is not a Happy number.

Example 7

Step 1: Consider number 7.

Step 2: Square the number => 72 => 7 x 7 => 49.

Step 3: Repeat the steps.

42 + 92 => 16 + 81 => 97

92 + 72 => 81 + 49 => 130

12 + 32 + 02 => 1 + 9 + 0 => 10

12 + 02 => 1 + 0 => 1

Repeating the steps gives the final number as 1. Therefore, the number considered in Step 1 is a Happy number.

Hope you have understood this mazing concept of the Happy number. To check whether a number is a Happy number or not, use the above examples. In case of any query, please feel free to provide your valuable comments.

Labels:

Comments

Post a Comment

Popular posts from this blog

The 3n + 1 Problem or Collatz Problem

The 3n + 1 Problem, also known as the 3n + 1 Conjecture or Collatz Problem or Collatz Conjecture, is that if any positive integer  n  is iteratively operated using two simple rules:  Addition  and  Division , then the end-result always leads to the number 1. The Rules Rule 1 [Addition]:  If the number n is odd, then triple it and add 1. Rule 2 [Division]:  If the number n is even, then divide it by 2. Steps to Follow Step 1:  Take any positive integer n. Step 2:  Check whether the integer is odd or even. Step 3:  If the integer is odd, then triple it and add 1. Step 4:  If the integer is even, then divide it by 2. Step 5:  Repeat this procedure. Step 6:  The end-result always leads to the number 1. For Example Integer n = 5 5 is an odd number, so triple it and add 1. The value of n = 16. 16 is an even number, so divide it by 2. The value of n = 8. 8 is an even number, so divide it by 2. The va...
2 comments
Read more

Universe Secret Code 369 Theory

When it comes to the origin of the universe secret code 369 theory, one name associated with this amazing concept is ‘Nikola Tesla’. Nikola Tesla [10 July 1856 – 07 January 1943] was not only a great inventor / scientist, but also the greatest mind of all the time. He has done a plenty of research throughout his life, and highly recognized for his contributions towards the following : High Voltage, High Frequency Power Experiments Alternating Current (AC) Electricity Supply System Apart from the above-mentioned contributions, he has given ‘Universe Secret Code 369 Theory’ and has called the number 369 – the key to the universe. Now the question that will come to the curious minds is that why Nikola Tesla has called this number the key to the universe. The answer to this question is described in this post. Let start with a specific pattern as shown below: 1 1 + 1 => 2 2 + 2 => 4 4 + 4 => 8 8 + 8 => 16; 1 + 6 => 7 16 + 16 => 32; 3 + 2 => 5 >>>>>>...
Post a Comment
Read more

Kaprekar Constant 6174 for Four-digit Number

Dattatreya Ramchandra Kaprekar (D. R. Kaprekar; 1905 – 1986) was an Indian mathematician who discovered several classes of natural numbers including Kaprekar Constant. The number 6174 is called Kaprekar Constant for four-digit number. Why is the number 6174 called Kaprekar Constant? In 1949, D. R. Kaprekar discovered an interesting property of the number 6174 through the mathematical calculations. D. R. Kaprekar observed that when any four-digit number of non-identical digits (not all digits are same) is arranged in two parts – highest number and lowest number, and difference is calculated using subtraction method, then the end-result will be always 6174 either in the first step or in no more than 7 steps. If the process is continued even after getting the end-result 6174, then the end-result will be always 6174 at each Iteration, therefore the number 6174 is known as Kaprekar Constant. Steps to Follow Take any four-digit number with different digits. Form the highest nu...
Post a Comment
Read more