202. Happy Number - Explanation

Description

A non-cyclical number is an integer defined by the following algorithm:

Given a positive integer, replace it with the sum of the squares of its digits.
Repeat the above step until the number equals 1, or it loops infinitely in a cycle which does not include 1.
If it stops at 1, then the number is a non-cyclical number.

Given a positive integer n, return true if it is a non-cyclical number, otherwise return false.

Example 1:

Input: n = 100

Output: true

Explanation: 1² + 0² + 0² = 1

Example 2:

Input: n = 101

Output: false

Explanation:
1² + 0² + 1² = 2
2² = 4
4² = 16
1² + 6² = 37
3² + 7² = 58
5² + 8² = 89
8² + 9² = 145
1² + 4² + 5² = 42
4² + 2² = 20
2² + 0² = 4 (This number has already been seen)

Constraints:

1 <= n <= 1000

Topics

Recommended Time & Space Complexity

You should aim for a solution as good or better than O(logn) time and O(logn) space, where n is the given integer.

Hint 1

Create a helper function that returns the sum of the squares of a number's digits. Then, simulate the given process. If we reach 1, return true. However, we may get stuck in a cycle if a number is processed more than once. What data structure can be used to detect if a number has already been processed?

Hint 2

We can use a hash set to detect if a number has already been processed. At each step, we update n with the return value of the helper function. If the result is 1, we return true. If n is already in the set, we return false. Otherwise, we add n to the hash set and continue.

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Prerequisites

Before attempting this problem, you should be comfortable with:

Hash Set - Detecting previously seen values to identify cycles
Floyd's Cycle Detection (Fast/Slow Pointers) - Detecting cycles without extra space
Digit Manipulation - Extracting digits using modulo and integer division

1. Hash Set

Intuition

A number is called happy if repeatedly replacing it with the sum of the squares of its digits eventually leads to 1.

While doing this process, only two things can happen:

we eventually reach 1 → the number is happy
we fall into a cycle and repeat numbers forever → the number is not happy

So the key problem is cycle detection.

A simple and beginner-friendly way to detect a cycle is to:

keep a set of numbers we have already seen
if a number repeats, we are stuck in a loop and will never reach 1

Algorithm

Initialize an empty set visit to store numbers we have already seen.
While n is not in visit:
- add n to visit
- replace n with the sum of the squares of its digits
- if n becomes 1, return true
If we exit the loop, it means n repeated:
- a cycle is detected
- return false

Helper Function (Sum of Squares)

To compute the next number:

Initialize output = 0
While n > 0:
- extract the last digit using n % 10
- square it and add to output
- remove the digit using n //= 10
Return output

class Solution:
    def isHappy(self, n: int) -> bool:
        visit = set()

        while n not in visit:
            visit.add(n)
            n = self.sumOfSquares(n)
            if n == 1:
                return True
        return False

    def sumOfSquares(self, n: int) -> int:
        output = 0

        while n:
            digit = n % 10
            digit = digit ** 2
            output += digit
            n = n // 10
        return output

public class Solution {
    public boolean isHappy(int n) {
        Set<Integer> visit = new HashSet<>();

        while (!visit.contains(n)) {
            visit.add(n);
            n = sumOfSquares(n);
            if (n == 1) {
                return true;
            }
        }
        return false;
    }

    private int sumOfSquares(int n) {
        int output = 0;

        while (n > 0) {
            int digit = n % 10;
            digit = digit * digit;
            output += digit;
            n /= 10;
        }
        return output;
    }
}

class Solution {
public:
    bool isHappy(int n) {
        unordered_set<int> visit;

        while (visit.find(n) == visit.end()) {
            visit.insert(n);
            n = sumOfSquares(n);
            if (n == 1) {
                return true;
            }
        }
        return false;
    }

private:
    int sumOfSquares(int n) {
        int output = 0;

        while (n > 0) {
            int digit = n % 10;
            digit = digit * digit;
            output += digit;
            n /= 10;
        }
        return output;
    }
};

class Solution {
    /**
     * @param {number} n
     * @return {boolean}
     */
    isHappy(n) {
        const visit = new Set();

        while (!visit.has(n)) {
            visit.add(n);
            n = this.sumOfSquares(n);
            if (n === 1) {
                return true;
            }
        }
        return false;
    }

    /**
     * @param {number} n
     * @return {number}
     */
    sumOfSquares(n) {
        let output = 0;

        while (n > 0) {
            let digit = n % 10;
            digit = digit * digit;
            output += digit;
            n = Math.floor(n / 10);
        }
        return output;
    }
}

public class Solution {
    public bool IsHappy(int n) {
        HashSet<int> visit = new HashSet<int>();

        while (!visit.Contains(n)) {
            visit.Add(n);
            n = SumOfSquares(n);
            if (n == 1) {
                return true;
            }
        }
        return false;
    }

    private int SumOfSquares(int n) {
        int output = 0;

        while (n > 0) {
            int digit = n % 10;
            digit = digit * digit;
            output += digit;
            n /= 10;
        }
        return output;
    }
}

func isHappy(n int) bool {
    visit := make(map[int]bool)

    for !visit[n] {
        visit[n] = true
        n = sumOfSquares(n)
        if n == 1 {
            return true
        }
    }
    return false
}

func sumOfSquares(n int) int {
    output := 0
    for n > 0 {
        digit := n % 10
        digit = digit * digit
        output += digit
        n = n / 10
    }
    return output
}

class Solution {
    fun isHappy(n: Int): Boolean {
        val visit = HashSet<Int>()

        var num = n
        while (num !in visit) {
            visit.add(num)
            num = sumOfSquares(num)
            if (num == 1) {
                return true
            }
        }
        return false
    }

    fun sumOfSquares(n: Int): Int {
        var output = 0
        var num = n
        while (num > 0) {
            val digit = num % 10
            output += digit * digit
            num /= 10
        }
        return output
    }
}

class Solution {
    func isHappy(_ n: Int) -> Bool {
        var visit = Set<Int>()
        var num = n

        while !visit.contains(num) {
            visit.insert(num)
            num = sumOfSquares(num)
            if num == 1 {
                return true
            }
        }
        return false
    }

    private func sumOfSquares(_ n: Int) -> Int {
        var num = n
        var output = 0

        while num > 0 {
            let digit = num % 10
            output += digit * digit
            num /= 10
        }
        return output
    }
}

Time & Space Complexity

Time complexity: $O(\log n)$
Space complexity: $O(\log n)$

2. Fast And Slow Pointers - I

Intuition

A number is happy if repeatedly replacing it with the sum of the squares of its digits eventually reaches 1.

Just like the hash set approach, the process can:

reach 1 → happy number
fall into a cycle → not a happy number

Instead of storing all visited numbers, we can detect a cycle using the fast and slow pointers technique (also known as Floyd's cycle detection).

The idea:

treat the transformation n → sumOfSquares(n) like moving through a linked list
use two pointers:
- slow moves one step at a time
- fast moves two steps at a time
if there is a cycle, slow and fast will eventually meet
if the cycle includes 1, then the number is happy

This avoids extra memory and still reliably detects cycles.

Algorithm

Initialize:
- slow = n
- fast = sumOfSquares(n)
While slow != fast:
- move slow one step:
  - slow = sumOfSquares(slow)
- move fast two steps:
  - fast = sumOfSquares(sumOfSquares(fast))
When the loop ends, a cycle is detected.
If fast == 1:
- the cycle ends at 1
- return true
Otherwise:
- the cycle does not include 1
- return false

Helper Function (Sum of Squares)

To compute the next number:

Initialize output = 0
While n > 0:
- extract the last digit using n % 10
- square it and add to output
- remove the digit using n //= 10
Return output

class Solution:
    def isHappy(self, n: int) -> bool:
        slow, fast = n, self.sumOfSquares(n)

        while slow != fast:
            fast = self.sumOfSquares(fast)
            fast = self.sumOfSquares(fast)
            slow = self.sumOfSquares(slow)
        return True if fast == 1 else False

    def sumOfSquares(self, n: int) -> int:
        output = 0

        while n:
            digit = n % 10
            digit = digit ** 2
            output += digit
            n = n // 10
        return output

public class Solution {
    public boolean isHappy(int n) {
        int slow = n, fast = sumOfSquares(n);

        while (slow != fast) {
            fast = sumOfSquares(fast);
            fast = sumOfSquares(fast);
            slow = sumOfSquares(slow);
        }

        return fast == 1;
    }

    private int sumOfSquares(int n) {
        int output = 0;
        while (n != 0) {
            output += (n % 10) * (n % 10);
            n /= 10;
        }
        return output;
    }
}

class Solution {
public:
    bool isHappy(int n) {
        int slow = n, fast = sumOfSquares(n);

        while (slow != fast) {
            fast = sumOfSquares(fast);
            fast = sumOfSquares(fast);
            slow = sumOfSquares(slow);
        }

        return fast == 1;
    }

private:
    int sumOfSquares(int n) {
        int output = 0;
        while (n != 0) {
            output += (n % 10) * (n % 10);
            n /= 10;
        }
        return output;
    }
};

class Solution {
    /**
     * @param {number} n
     * @return {boolean}
     */
    isHappy(n) {
        let slow = n;
        let fast = this.sumOfSquares(n);

        while (slow !== fast) {
            fast = this.sumOfSquares(fast);
            fast = this.sumOfSquares(fast);
            slow = this.sumOfSquares(slow);
        }

        return fast === 1;
    }

    /**
     * @param {number} n
     * @return {number}
     */
    sumOfSquares(n) {
        let output = 0;
        while (n !== 0) {
            output += (n % 10) ** 2;
            n = Math.floor(n / 10);
        }
        return output;
    }
}

public class Solution {
    public bool IsHappy(int n) {
        int slow = n, fast = sumOfSquares(n);

        while (slow != fast) {
            fast = sumOfSquares(fast);
            fast = sumOfSquares(fast);
            slow = sumOfSquares(slow);
        }

        return fast == 1;
    }

    private int sumOfSquares(int n) {
        int output = 0;
        while (n != 0) {
            output += (n % 10) * (n % 10);
            n /= 10;
        }
        return output;
    }
}

func isHappy(n int) bool {
    slow, fast := n, sumOfSquares(n)

    for slow != fast {
        fast = sumOfSquares(fast)
        fast = sumOfSquares(fast)
        slow = sumOfSquares(slow)
    }

    return fast == 1
}

func sumOfSquares(n int) int {
    output := 0
    for n > 0 {
        digit := n % 10
        output += digit * digit
        n = n / 10
    }
    return output
}

class Solution {
    fun isHappy(n: Int): Boolean {
        var slow = n
        var fast = sumOfSquares(n)

        while (slow != fast) {
            fast = sumOfSquares(fast)
            fast = sumOfSquares(fast)
            slow = sumOfSquares(slow)
        }

        return fast == 1
    }

    fun sumOfSquares(n: Int): Int {
        var output = 0
        var num = n
        while (num > 0) {
            val digit = num % 10
            output += digit * digit
            num /= 10
        }
        return output
    }
}

class Solution {
    func isHappy(_ n: Int) -> Bool {
        var slow = n
        var fast = sumOfSquares(n)

        while slow != fast {
            fast = sumOfSquares(fast)
            fast = sumOfSquares(fast)
            slow = sumOfSquares(slow)
        }
        return fast == 1
    }

    private func sumOfSquares(_ n: Int) -> Int {
        var num = n
        var output = 0

        while num > 0 {
            let digit = num % 10
            output += digit * digit
            num /= 10
        }
        return output
    }
}

Time & Space Complexity

Time complexity: $O(\log n)$
Space complexity: $O(1)$

3. Fast And Slow Pointers - II

Intuition

A number is happy if repeatedly replacing it with the sum of the squares of its digits eventually reaches 1.

Just like before, this process either:

reaches 1 → happy number
falls into a cycle → not a happy number

This solution uses a different cycle detection method called Brent's Algorithm, which is another form of fast–slow pointer technique.

Key idea:

We still move through the sequence n → sumOfSquares(n)
But instead of moving one pointer twice as fast every step, we:
- increase the distance between comparisons in powers of two
This reduces the number of comparisons and still guarantees cycle detection

We keep track of:

slow → a checkpoint value
fast → the moving value
power → how far we go before resetting slow
lam → current distance since last reset

If fast ever equals slow, a cycle is detected.

Algorithm

Initialize:
- slow = n
- fast = sumOfSquares(n)
- power = 1 (current block size)
- lam = 1 (steps taken in current block)
While slow != fast:
If power == lam:
- move the checkpoint:
  - slow = fast
- double the block size:
  - power *= 2
- reset step counter:
  - lam = 0
Move fast one step forward:
- fast = sumOfSquares(fast)
Increment lam by 1.
When the loop ends, a cycle is detected.
If fast == 1:
- the cycle ends at 1
- return true
Otherwise:
- return false

Helper Function (Sum of Squares)

To compute the next number:

Initialize output = 0
While n > 0:
- extract the last digit using n % 10
- square it and add to output
- remove the digit using n //= 10
Return output

class Solution:
    def isHappy(self, n: int) -> bool:
        slow, fast = n, self.sumOfSquares(n)
        power = lam = 1

        while slow != fast:
            if power == lam:
                slow = fast
                power *= 2
                lam = 0
            fast = self.sumOfSquares(fast)
            lam += 1
        return True if fast == 1 else False

    def sumOfSquares(self, n: int) -> int:
        output = 0

        while n:
            digit = n % 10
            digit = digit ** 2
            output += digit
            n = n // 10
        return output

public class Solution {
    public boolean isHappy(int n) {
        int slow = n, fast = sumOfSquares(n);
        int power = 1, lam = 1;
        while (slow != fast) {
            if (power == lam) {
                slow = fast;
                power *= 2;
                lam = 0;
            }
            lam++;
            fast = sumOfSquares(fast);
        }

        return fast == 1;
    }

    private int sumOfSquares(int n) {
        int output = 0;
        while (n != 0) {
            output += (n % 10) * (n % 10);
            n /= 10;
        }
        return output;
    }
}

class Solution {
public:
    bool isHappy(int n) {
        int slow = n, fast = sumOfSquares(n);
        int power = 1, lam = 1;

        while (slow != fast) {
            if (power == lam) {
                slow = fast;
                power *= 2;
                lam = 0;
            }
            lam++;
            fast = sumOfSquares(fast);
        }

        return fast == 1;
    }

private:
    int sumOfSquares(int n) {
        int output = 0;
        while (n != 0) {
            output += (n % 10) * (n % 10);
            n /= 10;
        }
        return output;
    }
};

class Solution {
    /**
     * @param {number} n
     * @return {boolean}
     */
    isHappy(n) {
        let slow = n;
        let fast = this.sumOfSquares(n);
        let power = 1,
            lam = 1;

        while (slow !== fast) {
            if (power === lam) {
                slow = fast;
                power *= 2;
                lam = 0;
            }
            lam++;
            fast = this.sumOfSquares(fast);
        }

        return fast === 1;
    }

    /**
     * @param {number} n
     * @return {number}
     */
    sumOfSquares(n) {
        let output = 0;
        while (n !== 0) {
            output += (n % 10) ** 2;
            n = Math.floor(n / 10);
        }
        return output;
    }
}

public class Solution {
    public bool IsHappy(int n) {
        int slow = n, fast = sumOfSquares(n);
        int power = 1, lam = 1;

        while (slow != fast) {
            if (power == lam) {
                slow = fast;
                power *= 2;
                lam = 0;
            }
            lam++;
            fast = sumOfSquares(fast);
        }

        return fast == 1;
    }

    private int sumOfSquares(int n) {
        int output = 0;
        while (n != 0) {
            output += (n % 10) * (n % 10);
            n /= 10;
        }
        return output;
    }
}

func isHappy(n int) bool {
    slow, fast := n, sumOfSquares(n)
    power, lam := 1, 1

    for slow != fast {
        if power == lam {
            slow = fast
            power *= 2
            lam = 0
        }
        fast = sumOfSquares(fast)
        lam++
    }

    return fast == 1
}

func sumOfSquares(n int) int {
    output := 0
    for n > 0 {
        digit := n % 10
        output += digit * digit
        n = n / 10
    }
    return output
}

class Solution {
    fun isHappy(n: Int): Boolean {
        var slow = n
        var fast = sumOfSquares(n)
        var power = 1
        var lam = 1

        while (slow != fast) {
            if (power == lam) {
                slow = fast
                power *= 2
                lam = 0
            }
            fast = sumOfSquares(fast)
            lam++
        }

        return fast == 1
    }

    fun sumOfSquares(n: Int): Int {
        var output = 0
        var num = n
        while (num > 0) {
            val digit = num % 10
            output += digit * digit
            num /= 10
        }
        return output
    }
}

class Solution {
    func isHappy(_ n: Int) -> Bool {
        var slow = n
        var fast = sumOfSquares(n)
        var power = 1
        var lam = 1

        while slow != fast {
            if power == lam {
                slow = fast
                power *= 2
                lam = 0
            }
            fast = sumOfSquares(fast)
            lam += 1
        }
        return fast == 1
    }

    private func sumOfSquares(_ n: Int) -> Int {
        var num = n
        var output = 0

        while num > 0 {
            let digit = num % 10
            output += digit * digit
            num /= 10
        }
        return output
    }
}

Time & Space Complexity

Time complexity: $O(\log n)$
Space complexity: $O(1)$

Common Pitfalls

Infinite Loop Without Cycle Detection

Forgetting to detect cycles causes the program to loop forever for non-happy numbers. Without a hash set or fast/slow pointers, numbers like 2 will endlessly cycle through the same values (2 → 4 → 16 → 37 → 58 → 89 → 145 → 42 → 20 → 4 → ...) without ever reaching 1.

Incorrect Digit Extraction

Using the wrong operations to extract digits leads to incorrect sum calculations. A common mistake is forgetting integer division when removing the last digit (n = n // 10) or using string conversion inefficiently. Always use n % 10 to get the last digit and n // 10 to remove it.