happy_numbers

Happy Number Checker

A tiny command‑line utility that determines whether a given integer is a happy number using Floyd’s cycle‑finding algorithm (the “tortoise and hare” technique). No external libraries—just the C standard library.

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1. Build & Run

chmod +x build.sh
./build
./happy_numbers 19   # prints that 19 is happy
./happy_numbers 20   # prints that 20 is not happy

Pass a single positive integer as argument. If omitted, the program prints a usage line.

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2. What’s a Happy Number?

A positive integer is happy if, after repeatedly replacing the number by the sum of the squares of its digits, you eventually reach 1. If you fall into a cycle that never reaches 1, the number is unhappy.

Example for 19:

1. 1² + 9² = 82
2. 8² + 2² = 68
3. 6² + 8² = 100
4. 1² + 0² + 0² = 1 → happy!

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3. Core Functions

3.1 sum_squared_digits()

int sum_squared_digits(int n) {
    int sum = 0;
    while (n > 0) {
        int d = n % 10;   // last digit
        sum += d * d;     // add square
        n /= 10;          // drop last digit
    }
    return sum;
}

Computes the “next” number in the happy‑number sequence.

3.2 Main Algorithm (Floyd’s Cycle Detection)

int slow = number;                   // tortoise: one step each loop
int fast = sum_squared_digits(number); // hare: two steps each loop

while (fast != 1 && slow != fast) {
    slow = sum_squared_digits(slow);
    fast = sum_squared_digits(sum_squared_digits(fast));
}

• If fast hits 1 → the sequence reaches 1 → happy.
• If slow == fast before 1 → we’re in a cycle that excludes 1 → unhappy.

Using two pointers avoids storing a set of seen values; memory cost is O(1).

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4. Output

• Happy: CONGRATULATION, <n> IS A HAPPY NUMBER!
• Unhappy: SORRY, <n> IS NOT A HAPPY NUMBER.

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5. Complexity

• Time: O(k·log₁₀ n) where k is the number of iterations until cycle detection—bounded because sums shrink quickly.
• Memory: O(1) thanks to Floyd’s algorithm.