Sad Number in Java

A Sad Number (also known as an Unhappy Number) is a number that never reaches 1 when we keep replacing it with the sum of the squares of its digits. Instead, it enters a repeating loop forever.

Before writing the Java program, let's deeply understand how Sad Numbers behave. We’ll walk step-by-step, ask interesting questions, and prove why certain assumptions fail.

Example: Step-by-Step Dry Run

• Example: 4
  Steps:
  4² = 16
  1² + 6² = 37
  3² + 7² = 58
  5² + 8² = 89
  8² + 9² = 145
  1² + 4² + 5² = 42
  4² + 2² = 20
  2² + 0² = 4 (Loop repeats)
  
  ✔️ 4 is a Sad Number

A Curious Thought… 🤔

Looking at the example above, you might wonder:

This sounds logical at first… but unfortunately, it does not work in most cases.

Let’s Prove This Wrong

Consider another number:

• Example: 12
  Steps:
  1² + 2² = 5
  5² = 25
  2² + 5² = 29
  2² + 9² = 85
  8² + 5² = 89
  → From here, the cycle becomes:
  89 → 145 → 42 → 20 → 4 → 16 → 37 → 58 → 89 → ...
  
  The important part:
  12 never appears again!
  
  ✔️ 12 is still a Sad Number

So the loop does not care about the original number. It only enters the fixed unhappy cycle.

The Universal Sad Number Loop

All Sad Numbers eventually fall into this 8-number loop:

So checking whether the original number returns will fail for almost all values.

So How Do We Detect a Sad Number Correctly?

We simply need to detect when a loop begins. And to do that, we need to track numbers we have already seen.

Enter HashSet — Our Loop Detector

A HashSet in Java is a data structure that:

• Stores unique values
• Allows fast lookup (O(1))
• Helps us detect repetition instantly

If a number appears again → a loop has started → the number is Sad.

Java Program to Check Sad Number

import java.util.HashSet;

public class SadNumber {

    public static int digitSquareSum(int num) {
        int sum = 0;
        while (num > 0) {
            int digit = num % 10;
            sum += digit * digit;
            num /= 10;
        }
        return sum;
    }

    public static void main(String[] args) {
        int num = 12; // Change this to test other numbers

        HashSet seen = new HashSet<>();

        while (num != 1 && !seen.contains(num)) {
            seen.add(num);
            num = digitSquareSum(num);
        }

        if (num == 1) {
            System.out.println("Not a Sad Number (It is a Happy Number)");
        } else {
            System.out.println("Sad Number");
        }
    }
}
    

Practice Challenges