Is 9 a happy number?

No, 9 is not a happy number.

A happy number is defined as a number which eventually reaches 1 when replaced by the sum of the square of each digit. If this process leads to a cycle that does not include 1, the number is considered unhappy.

Let's trace the sequence for the number 9:

1. 9² = 81
2. 8² + 1² = 64 + 1 = 65
3. 6² + 5² = 36 + 25 = 61
4. 6² + 1² = 36 + 1 = 37
5. 3² + 7² = 9 + 49 = 58
6. 5² + 8² = 25 + 64 = 89
7. 8² + 9² = 64 + 81 = 145
8. 1² + 4² + 5² = 1 + 16 + 25 = 42
9. 4² + 2² = 16 + 4 = 20
10. 2² + 0² = 4 + 0 = 4
11. 4² = 16
12. 1² + 6² = 1 + 36 = 37

Notice that we've encountered 37 again, indicating that the sequence will cycle endlessly and never reach 1. Specifically, unhappy numbers often end up in the cycle: 4, 16, 37, 58, 89, 145, 42, 20, 4...

Therefore, since 9 enters a cycle that does not include 1, it is an unhappy number.