Bulb Toggle II

What this hard puzzle question tests

This is a hard combinatorial puzzle that requires you to track state changes across many iterations and find a pattern or closed-form insight rather than brute-force simulation. It appears in quant interview loops where firms want to see whether you can reason systematically about complex, repetitive processes and extract structure from apparent chaos.

The key is recognizing which bulbs end up toggled an odd number of times (and thus remain on) after all 1011 people have acted. This demands careful bookkeeping of divisibility relationships and, ideally, a mathematical shortcut that avoids computing all 2021 individual bulb states by hand.

• Divisibility and factorization patterns
• Parity reasoning (odd vs. even counts)
• Finding closed-form or recursive structure in iterative processes