Happy Numbers

                                                                                         

Introduction

A ‘happy number’ is a special type of number.
A "happy number" is defined by the following process.

1. Take any positive integer ‘p’.
2. Break it down into digits.
3. Square the digits and add those squares.
4. You get a new number ‘n’.
5. If n = 1, then ‘p’ is a happy number; if not, set p = n and repeat the above steps.
6. If the process repeats forever, then ‘p’ is not a happy number.

Let’s look at a simple example.

7 is a happy number.
Let me take you through the step-by-step process.

Iteration 1

7 is a single digit number.
Square it…we get, 7^2 = 49.
49 is NOT equal to 1. So, let’s repeat the process.

Iteration 2

49 has two digits 4 and 9.
Square them…we get 4^2 = 16 and 9^2 = 81.

Add the squares.
16 + 81 = 97

97 is NOT equal to 1. So, repeat the process.

9^2 + 7^2 = 130 (Iteration 3)
1^2 + 3^2 + 0^2 = 10 (Iteration 4)
1^2 + 0^2 = 1 ---> We ended up at 1. (Iteration 5)
So, 7 is a happy number.

It took 5 iterations to conclude that ‘7’ is a happy number.
You may be surprised to find that it takes only a few steps to conclude that a bigger number like 1,122 is happy!

1,122 has four digits 1, 1, 2, and 2.

Add their squares. We get:
1^2 + 1^2 + 2^2 + 2^2 = 1 + 1 + 4 + 4 = 10

Repeat the process.
1^2 + 0^2 = 1 + 0 = 1

The following are the HAPPY NUMBERS up to 100.
1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100

How to identify unhappy numbers?
Numbers that are not happy are called unhappy numbers.

If a number is not happy then the iteration process discussed above never ends at 1. Instead, the process enters into an infinite loop.
The loop is this:
4, 16, 37, 58, 89, 145, 42, 20, 4, 16, 37, 58, ---> we can stop the iteration process once we get any one of these numbers.
For example:
Let’s check out if 124 is a happy number.

1^2 + 2^2 + 4^2 = 21
2^2 + 1^2 = 5
5^2 = 25
2^2 + 5^2 = 29
2^2 + 9^2 = 85
8^2 + 5^2 = 89

We’ve reached 89! We can stop the iteration process here and conclude that 124 is not a happy number. If we proceed further, we’ll get into an infinite loop and will never get out of it.


Some interesting facts about “Happy Numbers”:

(1) All powers of 10 are happy numbers.
(2) Reversing the digits of a HAPPY NUMBER produces another HAPPY NUMBER.

For example, 19 is a happy number, 91 is happy as well. Try it!

(3) Multiplying a happy number by any power of ten yields another happy number.
Let’s consider the same example.
19 is a happy number…so are 190, 1900, 19000 and so on.

(4) By rearranging the digits of a happy number, we get another happy number.
Consider the same example we’ve just discussed.
19, 190, 1900, 19000 are all happy numbers.

Let me rearrange the digits.
91, 109, 1009, 1090, 10009, 10900, 10090, 91000 are also happy numbers.

About the author:
I’m Chandrajeet, an in-house writer for iCoachMath. iCoachMath is an effective, convenient, easy-to-use online Math Program which has been used by thousands of students, teachers, and parents.iCoachMath strives to lead K-12 students to excellence in math by offering quality web-based educational solutions. iCoachMath’s instructional and lesson materials are aligned to State Curriculum Standards in all 50 states (USA). iCoachMath

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