diff --git a/dsa-solutions/gfg-solutions/0004-fibbonacci-sum.md b/dsa-solutions/gfg-solutions/0004-fibbonacci-sum.md index a33924e21..6fc2b34d3 100644 --- a/dsa-solutions/gfg-solutions/0004-fibbonacci-sum.md +++ b/dsa-solutions/gfg-solutions/0004-fibbonacci-sum.md @@ -19,8 +19,8 @@ Given a positive number N, find the value of $f0 + f1 + f2 + ... + fN$ where fi ## Examples **Example 1:** -``` +``` Input: N = 3 Output: @@ -30,16 +30,14 @@ Explanation: ``` **Example 2:** -``` - +``` Input: N = 4 Output: 7 Explanation: 0 + 1 + 1 + 2 + 3 = 7 - ``` ## Your Task @@ -51,7 +49,7 @@ Expected Auxiliary Space: $O(1)$ ## Constraints -1. $1 \leq N \leq 100000$ +- $(1 \leq N \leq 100000)$ ## Solution Approach @@ -66,7 +64,7 @@ We can compute the sum of Fibonacci numbers from f0 to fN using a simple iterati 1. Initialize `sum` to 0. 2. Use a loop to compute Fibonacci numbers up to N. 3. Add each Fibonacci number to `sum`. -4. Return the sum modulo 1000000007. +4. Return the sum modulo $1000000007$. #### Code (C++): @@ -267,4 +265,7 @@ By leveraging the properties of Fibonacci numbers and matrix multiplication, we - **GeeksforGeeks Problem:** [Geeks for Geeks Problem](https://www.geeksforgeeks.org/problems/fibonacci-sum/0) - **Solution Link:** [Fibonacci Sum on Geeks for Geeks](https://www.geeksforgeeks.org/problems/fibonacci-sum/0) - **Authors GeeksforGeeks Profile:** [Vipul](https://www.geeksforgeeks.org/user/lakumvipwjge/) +<<<<<<<<< Temporary merge branch 1 +========= +>>>>>>>>> Temporary merge branch 2 diff --git a/dsa-solutions/gfg-solutions/0006-next-happy-number.md b/dsa-solutions/gfg-solutions/0006-next-happy-number.md new file mode 100644 index 000000000..afd83100d --- /dev/null +++ b/dsa-solutions/gfg-solutions/0006-next-happy-number.md @@ -0,0 +1,259 @@ +--- +id: next-happy-number +title: Next Happy Number (Geeks for Geeks) +sidebar_label: 0004 - Next Happy Number +tags: + - intermediate + - Fibonacci + - Dynamic Programming + - Mathematics + - Algorithms +--- + +This tutorial contains a complete walk-through of the Next Happy Number problem from the Geeks for Geeks website. It features the implementation of the solution code in three programming languages: Python, C++, and Java. + +## Problem Statement + +For a given non-negative integer N, find the next smallest Happy Number. A number is called Happy if it leads to 1 after a sequence of steps, where at each step the number is replaced by the sum of squares of its digits. If we start with a Happy Number and keep replacing it with the sum of squares of its digits, we eventually reach 1. + +## Examples + +**Example 1:** + +``` +input: +N = 8 + +Output: +10 +``` + +Explanation: +Next happy number after 8 is 10 since +$[ 1 * 1 + 0 * 0 = 1]$ + +**Example 2** + +``` +Input: +N = 10 + +Output: +13 +``` + +Explanation: + +After 10, 13 is the smallest happy number because +$[1 * 1 + 3 * 3 = 10]$, so we replace 13 by 10 and $[1 * 1 + 0 * 0 = 1]$. + +## Task + +You don't need to read input or print anything. Your task is to complete the function `nextHappy()` which takes an integer N as input parameters and returns an integer, the next Happy number after N. + +## Constraints + +- $(1 \leq N \leq 10^5)$ + + +## Solution Approach + +### Intuition + +To solve the problem, we need to: + +1. Identify the next number greater than N. +2. Check if it is a Happy Number. +3. Repeat the process until we find the next Happy Number. + +A number is identified as Happy if, by repeatedly replacing it with the sum of squares of its digits, we eventually reach 1. + +### Steps + +1. Implement a helper function to determine if a number is Happy. +2. Start checking numbers greater than N, using the helper function to identify the next Happy Number. + +### Detailed Explanation + +The numbers that, when you repeatedly sum the squares of their digits, eventually result in 1 are known as "happy numbers." + +Here are examples of how to determine if numbers less than 10 are happy numbers: + +- **Number 1:** + + $[1 ^ 2 = 1]$ + Since we have already reached 1, the process stops here. 1 is a happy number. + +- **Number 2:** + + $[2^2 = 4]$ + $[4^2 = 16]$ + $[1^2 + 6^2 = 37]$ + $[3^2 + 7^2 = 58]$ + $[5^2 + 8^2 = 89]$ + $[8^2 + 9^2 = 145]$ + $[1^2 + 4^2 + 5^2 = 42]$ + $[4^2 + 2^2 = 20]$ + $[2^2 + 0^2 = 4]$ + + Since we have reached 4 again, the process will continue in an infinite loop. 2 is not a happy number. + +- **Number 3:** + + Similar to the above steps, 3 will also enter a loop and is not a happy number. + +- **Number 4:** + + Similar to the above steps, 4 will also enter a loop and is not a happy number. + +- **Number 5:** + + Similar to the above steps, 5 will also enter a loop and is not a happy number. + +- **Number 6:** + + Similar to the above steps, 6 will also enter a loop and is not a happy number. + +- **Number 7:** + + $[7^2 = 49]$ + $[4^2 + 9^2 = 97]$ + $[9^2 + 7^2 = 130]$ + $[1^2 + 3^2 + 0^2 = 10]$ + $[1^2 + 0^2 = 1]$ + + Since we have reached 1, the process stops here. 7 is a happy number. + +- **Number 8:** + + Similar to the above steps, 8 will also enter a loop and is not a happy number. + +- **Number 9:** + + Similar to the above steps, 9 will also enter a loop and is not a happy number. + +Based on this analysis, the numbers less than 10 that result in 1 when you repeatedly sum the squares of their digits are: 1 and 7. + +### Implementation + +#### Code (C++): + +```cpp +#include + +class Solution { +public: + bool solve(int n) { + if (n == 1 || n == 7) return true; + if (n == 2 || n == 4 || n == 8 || n == 3 || n == 9 || n == 5 || n == 6) return false; + int sq_sum = 0; + while (n) { + int x = n % 10; + sq_sum += (x * x); + n /= 10; + } + return solve(sq_sum); + } + + int nextHappy(int n) { + while (true) { + n++; + if (solve(n)) return n; + } + } +}; + +int main() { + Solution sol; + int N = 8; + std::cout << "Next happy number after " << N << " is: " << sol.nextHappy(N) << std::endl; + return 0; +} + +``` + +#### Code(Python) + +```python +class Solution: + def solve(self, n: int) -> bool: + if n == 1 or n == 7: + return True + if n in {2, 4, 8, 3, 9, 5, 6}: + return False + sq_sum = 0 + while n > 0: + x = n % 10 + sq_sum += (x * x) + n //= 10 + return self.solve(sq_sum) + + def nextHappy(self, n: int) -> int: + while True: + n += 1 + if self.solve(n): + return n + +# Example usage +sol = Solution() +N = 8 +print(f"Next happy number after {N} is: {sol.nextHappy(N)}") + +``` + +#### Code (Java) + +```java +public class Solution { + public boolean solve(int n) { + if (n == 1 || n == 7) return true; + if (n == 2 || n == 4 || n == 8 || n == 3 || n == 9 || n == 5 || n == 6) return false; + int sq_sum = 0; + while (n > 0) { + int x = n % 10; + sq_sum += (x * x); + n /= 10; + } + return solve(sq_sum); + } + + public int nextHappy(int n) { + while (true) { + n++; + if (solve(n)) return n; + } + } + + public static void main(String[] args) { + Solution sol = new Solution(); + int N = 8; + System.out.println("Next happy number after " + N + " is: " + sol.nextHappy(N)); + } +} + +``` + +### Complexity + +- **Time Complexity:** $O(klog_{10}N)$ due to the operations on digits of the numbers. + +#### Explanation : + +- We will be able to determine whether a number is happy or not after recursively calling the solve function and adding the square of its digits for a maximum of 15-20 times (you can check for any random value less than 10^5). Let's denote this value as x. +- The time taken to add the square of the digits is log10(n). Furthermore, as we are checking until we find the happy number, let's denote the number of iterations as c = (happy number>n) -n +- Let's denote x\*c=k; + Therefore, TC: (klog10(n)) +- The value of k won't even reach 10^4 or 10^5. You can try this approach with any random value. + +- **Space Complexity:** $O(1)$ since we only use a fixed amount of extra space for the set to store seen numbers. + +## Conclusion + +To find the next Happy Number after a given integer N, we can implement a solution that iteratively checks each number greater than N until a Happy Number is found. This solution efficiently identifies Happy Numbers using a helper function to compute the sum of squares of digits and a set to track previously seen numbers to avoid infinite loops. + +## References + +- **GeeksforGeeks Problem:** [Geeks for Geeks Problem](https://www.geeksforgeeks.org/problems/fibonacci-sum/0) +- **Solution Link:** [Fibonacci Sum on Geeks for Geeks](https://www.geeksforgeeks.org/problems/fibonacci-sum/0) +- **Authors GeeksforGeeks Profile:** [Vipul](https://www.geeksforgeeks.org/user/lakumvipwjge/) +