Is 9 an Unhappy Number?

Yes, 9 is an unhappy number.

According to the provided reference, "The first few unhappy numbers are 2, 3, 4, 5, 6, 8, 9, 11, 12, 14, 15, 16, 17, 18, 20, ...", which explicitly includes 9 in the list of unhappy numbers.

Understanding Unhappy Numbers

An unhappy number is defined by a specific process:

1. Take a positive integer.
2. Square each of its digits.
3. Sum the results.
4. Repeat steps 2 and 3 with the new sum.
5. Continue this process until either the sum is 1 (in which case, the original number is considered a happy number) or the sequence enters a cycle that never reaches 1 (in which case, the original number is considered an unhappy number).

Why 9 Is Unhappy

Let's apply this process to 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

As you can see, the process leads back to 37, and thus a repeating sequence (37, 58, 89, 145, 42, 20, 4, 16) will continue indefinitely and never reaches 1. This confirms that 9 is indeed an unhappy number.

Step	Calculation	Result
1	9²	81
2	8² + 1²	65
3	6² + 5²	61
4	6² + 1²	37
5	3² + 7²	58
6	5² + 8²	89
7	8² + 9²	145
8	1² + 4² + 5²	42
9	4² + 2²	20
10	2² + 0²	4
11	4²	16
12	1² + 6²	37