Solution next happy number added

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

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 <iostream>
+
+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/)
+