Happy Number Calculator

Welcome to the Happy Number Calculator, an interactive tool to explore one of the most delightful concepts in recreational mathematics. Enter any positive integer and discover whether it is a happy number by watching its digit-squaring journey unfold step by step. The animated visualization, trajectory chart, and detailed breakdown make it easy to understand how numbers reach happiness — or fall into an endless cycle.

What is a Happy Number?

A happy number is a positive integer that eventually reaches 1 when you repeatedly replace it with the sum of the squares of its digits. If a number never reaches 1 and instead enters an infinite cycle, it is called an unhappy number (or sad number).

The Process

Where \(d_i\) are the individual digits of \(n\). For example:

• 19: 1² + 9² = 1 + 81 = 82 → 8² + 2² = 64 + 4 = 68 → 6² + 8² = 36 + 64 = 100 → 1² + 0² + 0² = 1 ✔ Happy!
• 2: 2² = 4 → 4² = 16 → 1² + 6² = 37 → ... → enters cycle ✘ Unhappy

The Unhappy Cycle

Every unhappy number eventually enters the same 8-number cycle:

This remarkable fact means you only need to check if the sequence ever reaches 4 to determine that a number is unhappy. There is no other cycle for the digit-squaring process in base 10.

How to Use This Calculator

1. Enter a number: Type any positive integer into the input field. Try the quick examples for classic happy and unhappy numbers.
2. Choose a mode: Use "Check Single Number" to analyze one number in depth, or "Find All Happy Numbers in Range" to discover every happy number from 1 to N.
3. Analyze results: Click "Check Number" to see the complete digit-squaring sequence, interactive trajectory chart, and step-by-step breakdown.
4. Explore patterns: Try different numbers to discover relationships between happy numbers and their digit patterns.

Happy Numbers Under 100

There are exactly 20 happy numbers between 1 and 100:

Interesting Properties of Happy Numbers

Digit Permutation Invariance

If a number is happy, then any rearrangement of its digits is also happy. For example, since 19 is happy, 91 is also happy. Similarly, 13, 31 are both happy.

Zero Insertion Invariance

Inserting or removing zeros does not change whether a number is happy. Since 19 is happy, 109, 190, 1009, 1090 are all happy too.

Happy Primes

A happy prime is a number that is both happy and prime. The first few happy primes are: 7, 13, 19, 23, 31, 79, 97, 103, 109, 139. Happy primes have attracted interest in number theory and have even been referenced in popular culture.

Density of Happy Numbers

The density of happy numbers among all positive integers is approximately 14.3%. This means roughly 1 in 7 positive integers is happy. Interestingly, this density remains fairly consistent across different ranges.

Happy Numbers in Different Bases

The concept of happy numbers can be extended to other number bases. In base \(b\), a number is happy if the iterated sum of squares of its base-\(b\) digits reaches 1. The properties and cycle structures vary significantly between bases:

• Base 2: Only powers of 2 are happy (1, 2, 4, 8, 16, ...)
• Base 4: 1 is the only happy number
• Base 10: The familiar set with approximately 14.3% density

Computational Aspects

For any number with \(d\) digits, the sum of squares of its digits is at most \(81d\) (when all digits are 9). This means:

• A 1-digit number maps to at most 81
• A 3-digit number (up to 999) maps to at most 243
• Any number above 999 immediately maps to a smaller number

This guarantees that the sequence always eventually enters a cycle or reaches 1, making the algorithm always terminate.

Frequently Asked Questions

What is a happy number?

A happy number is a positive integer that eventually reaches 1 when you repeatedly replace it with the sum of the squares of its digits. For example, 19 is happy because: 1² + 9² = 82, then 8² + 2² = 68, then 6² + 8² = 100, then 1² + 0² + 0² = 1. Numbers that never reach 1 are called unhappy or sad numbers.

What happens to unhappy numbers?

Unhappy (or sad) numbers never reach 1. Instead, they eventually enter an infinite cycle: 4 → 16 → 37 → 58 → 89 → 145 → 42 → 20 → 4. Every number that is not happy will eventually enter this exact 8-number cycle.

What are all the happy numbers under 100?

The happy numbers under 100 are: 1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97. That is 20 happy numbers out of 100, or exactly 20%.

Is there a pattern to happy numbers?

Happy numbers share interesting properties: any permutation of the digits of a happy number is also happy (e.g., 19 and 91 are both happy). Additionally, inserting or removing zeros does not change happiness (e.g., 19, 109, 190 are all happy). The density of happy numbers is approximately 14.3% among all positive integers.

Who discovered happy numbers?

Happy numbers were first studied by Reg Allenby in 1966. The concept gained popularity through recreational mathematics. The term "happy number" is attributed to a young student who brought the idea to mathematicians at Cambridge.

How are happy numbers used in mathematics?

Happy numbers appear in number theory, recreational mathematics, and are used as programming exercises. They connect to concepts like fixed points, cycles in iterated functions, and digit-based sequences. Happy primes (numbers that are both happy and prime) are of particular interest in mathematical research.

Additional Resources

• Happy Number - Wikipedia
• OEIS A007770 - Happy Numbers