Bulb Toggle I

What this toggle-sequence puzzle tests

This is a medium-difficulty puzzle that appears in quant interviews to assess pattern recognition and mathematical reasoning under a simple rule. The question asks you to trace the outcome of a systematic sequence of operations without explicitly computing all steps—a useful proxy for whether you can identify structure in seemingly complex processes.

The key insight is realizing that you don't need to simulate all 1,000,000 toggles. Instead, you should think about what determines the final state of each individual bulb: how many times does it get toggled, and what does that parity tell you? Once you frame it that way, the problem reduces to a number-theoretic property that applies consistently across all bulbs.

• Parity and even/odd reasoning
• Divisibility and factors
• Reducing brute-force simulation to closed-form insight