Bit Manipulation Based

Bit manipulation is appropriate only when the constraints imply a binary invariant. Ask whether numbers fit a fixed width, whether duplicates follow a known frequency pattern, and whether ordering matters. XOR is useful only when values occur in cancelling pairs; for counts modulo three or another frequency, maintain per-bit counts or a finite-state bitmask approach. A mask such as one shifted left by i represents membership of item i, so it is practical only when the universe is small enough for the word or a bitset. For range operations, distinguish signed arithmetic from unsigned bit patterns and be explicit about language behavior. Test zero, a single high bit, negative inputs if allowed, repeated values that violate the stated pattern, and overflow around shifts. Typical per-value bit processing is O(n times W) for W-bit integers, which is O(n) for a fixed machine width, and uses O(1) extra space. Do not claim a generic problem is solved by XOR before proving cancellation.