Bit Manipulation

Bit manipulation uses the binary representation of integers directly through bitwise operators: AND, OR, XOR, and shifts. It is useful for problems involving pairs, uniqueness, flags, or any situation where you need to extract or toggle individual bits. XOR is particularly powerful because it is its own inverse: XOR-ing a value twice cancels out. The problems here are mostly short, but they require a different way of thinking about numbers. Once you internalize the basic bit operations, you will start seeing where they can replace more expensive data structures in problems across other categories.