Bulb Switcher II

Bulb Switcher II

What is this problem about?

The Bulb Switcher II interview question is an advanced version of the original. You have n bulbs and a set of 4 buttons that toggle different subsets of bulbs (all, evens, odds, or 3k+1). After exactly m button presses, you need to find the number of unique configurations the bulbs can be in. This Bulb Switcher II coding problem requires identifying symmetries and periodicity.

Why is this asked in interviews?

Microsoft uses this to test a candidate's ability to simplify a large state space. While it looks like a BFS/DFS problem, the key is realizing that the bulbs' states repeat every 6 bulbs. Furthermore, after a few button presses, the number of possible states stabilizes. It's a test of whether you can reduce a potentially infinite search space into a few manageable cases.

Algorithmic pattern used

This follows the Math, Breadth-First Search, Bit Manipulation interview pattern. However, the optimal solution is a Case-Based Analysis. You identify that the state of any bulb is determined by its index modulo 6. This reduces the effective n to at most 6. Then, you analyze how many states can be reached based on m (1, 2, or 3+ presses).

Example explanation

If n = 1 and m = 1:

• Any of the 4 buttons will either keep the bulb ON or turn it OFF.
• Possible states: [ON], [OFF]. Result: 2. As n and m increase, the number of combinations grows, but it caps out at 8 unique configurations because of the mathematical relationships between the four operations (e.g., toggling all + toggling evens = toggling odds).

Common mistakes candidates make

• Exhaustive BFS: Trying to simulate every possible sequence of m button presses. Since each step has 4 choices, the complexity is O(4^M), which is exponential.
• Ignoring periodicity: Not realizing that the state of bulb 7 is always tied to bulb 1, and bulb 8 to bulb 2, etc.
• Handling small N/M: Getting the edge cases wrong when n or m are very small (like 0 or 1).

Interview preparation tip

Practice "state space reduction." When a problem has a huge number of operations but only a few possible outcomes, try to find the "period" or the point where the behavior starts to repeat.