feat: add Sum of Two Integers explanation

- Bit manipulation algorithm approach
- Comprehensive problem analysis
- Multiple approaches and optimizations
- Edge cases and applications

Author: TechnoBlogger14o3

neetcode-150/math-geometry/sum-of-two-integers/explanation.md

@@ -0,0 +1,76 @@
+# Sum of Two Integers
+
+## Problem Statement
+
+Given two integers `a` and `b`, return the sum of the two integers without using the operators `+` and `-`.
+
+## Examples
+
+**Example 1:**
+```
+Input: a = 1, b = 2
+Output: 3
+```
+
+## Approach
+
+### Method 1: Bit Manipulation (Recommended)
+1. Use XOR for sum without carry
+2. Use AND and left shift for carry
+3. Repeat until no carry
+4. Most efficient approach
+
+**Time Complexity:** O(32) - Constant time
+**Space Complexity:** O(1) - No extra space
+
+### Method 2: Iterative Addition
+1. Use bit manipulation iteratively
+2. Less efficient than recursive approach
+
+**Time Complexity:** O(32) - Constant time
+**Space Complexity:** O(1) - No extra space
+
+## Algorithm
+
+```
+1. While b != 0:
+   a. sum = a ^ b (sum without carry)
+   b. carry = (a & b) << 1 (carry)
+   c. a = sum, b = carry
+2. Return a
+```
+
+## Key Insights
+
+- **Bit Manipulation**: Use XOR and AND operations
+- **Local Optimum**: Calculate sum and carry efficiently
+- **Global Optimum**: Complete addition without + and -
+- **Space Optimization**: Use only necessary space
+
+## Alternative Approaches
+
+1. **Iterative**: Use iterative approach
+2. **Recursive**: Use recursive approach
+3. **Lookup Table**: Use precomputed table
+
+## Edge Cases
+
+- Zero: Handle appropriately
+- Negative numbers: Handle two's complement
+- Large numbers: Handle efficiently
+- Same numbers: Handle appropriately
+
+## Applications
+
+- Bit manipulation
+- Arithmetic operations
+- Algorithm design patterns
+- Interview preparation
+- System design
+
+## Optimization Opportunities
+
+- **Bit Manipulation**: Most efficient approach
+- **Space Optimization**: O(1) space complexity
+- **Constant Time**: O(32) time complexity
+- **No Extra Space**: Use only necessary space