Calculator Form
Enter a single integer, several integers, or a range. The checker evaluates digit-square iterations and identifies happy, unhappy, or limited outcomes.
Example Data Table
These sample values illustrate common outcomes in base 10 with a practical iteration limit.
| Number | Base | Status | Steps | Ending detail |
|---|---|---|---|---|
| 1 | 10 | Happy | 0 | Already at 1 |
| 7 | 10 | Happy | 5 | 7 → 49 → 97 → 130 → 10 → 1 |
| 19 | 10 | Happy | 4 | 19 → 82 → 68 → 100 → 1 |
| 20 | 10 | Unhappy | 8 | Returns to 20 through a repeating loop |
| 116 | 10 | Unhappy | 11 | Falls into the standard unhappy cycle |
Formula Used
Happy number transform: Hb(n) = Σ di2
Write the integer in base b, square each digit, and add those squares. Repeat the transform on each new result.
If the process reaches 1, the number is happy. If a previous state repeats before reaching 1, the number is unhappy.
This calculator also supports an iteration cap. That cap helps analyze long runs without overextending batch evaluations.
How to Use This Calculator
1. Enter one integer, many integers, or a numeric range.
2. Choose the working base from 2 through 16.
3. Set the maximum iteration count for stopping control.
4. Enable the detailed table when you want step-by-step expressions.
5. Press submit to view the summary, graph, and result table above the form.
6. Use the export buttons to download the current result set as CSV or PDF.
FAQs
1. What is a happy number?
A happy number eventually reaches 1 after repeatedly replacing the number with the sum of the squares of its digits. If it never reaches 1 and starts repeating earlier values, it is unhappy.
2. Why does the calculator allow different bases?
The digit-square process depends on how the number is written. Changing the base changes the digits, so the transformation path can change as well. This is useful for mathematical exploration and teaching.
3. What does an unhappy result mean?
An unhappy result means the sequence entered a loop before reaching 1. Once a previous value appears again, every later step repeats the same cycle, so the number cannot become happy.
4. What is the iteration limit for?
The iteration limit is a safety boundary. It prevents extremely long exploratory runs in large batches. If the limit is hit before reaching 1 or finding a repeated value, the result is marked limited.
5. Can I test multiple numbers at once?
Yes. Enter values separated by commas, spaces, or line breaks. You can also define a numeric range with a custom step. The calculator merges both sources into one evaluated batch.
6. What does the graph show?
For a single value, the graph shows each transformation result across iterations. For batch runs, it compares how many steps each tested number needs to reach 1, repeat a cycle, or stop at the limit.
7. Why is 1 always happy?
Because it is already the target state. Applying the rule to 1 keeps the process at 1, so no additional work is needed. That makes it the simplest happy number.
8. Are zero and negative numbers supported?
This checker accepts non-negative integers only. Traditional happy-number work usually focuses on positive integers. Zero is allowed for experimentation, but it does not reach 1 under the standard digit-square process.