Happy Number Calculator

Understanding Happy Numbers

What is a Happy Number?

A Happy Number is a positive integer that, when you repeatedly replace it with the sum of the squares of its digits, eventually reaches 1. If this process results in an endless cycle that does not include 1, then the number is called a Sad Number.

Happy Number Formula

For a number \(n\), the process to determine if it's a Happy Number is:

\[n \rightarrow \sum_{i} d_i^2\]

Where \(d_i\) are the digits of \(n\).

Repeat this process until either:

• The result becomes 1 (Happy Number)
• The process enters a cycle that doesn't include 1 (Sad Number)

Calculation Steps

Let's determine if 19 is a Happy Number:

1. \(19 \rightarrow 1^2 + 9^2 = 1 + 81 = 82\)
2. \(82 \rightarrow 8^2 + 2^2 = 64 + 4 = 68\)
3. \(68 \rightarrow 6^2 + 8^2 = 36 + 64 = 100\)
4. \(100 \rightarrow 1^2 + 0^2 + 0^2 = 1 + 0 + 0 = 1\)

Since we reached 1, 19 is a Happy Number.

Example and Visual Representation

This diagram illustrates that 19 is a Happy Number, showing the sequence that leads to 1.