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Bulb Switcher II

Difficulty: Medium

Problem Description

There is a room with n bulbs labeled from 1 to n that all are turned on initially, and four buttons on the wall. Each of the four buttons has a different functionality where:

  • Button 1: Flips the status of all the bulbs.
  • Button 2: Flips the status of all the bulbs with even labels (i.e., 2, 4, ...).
  • Button 3: Flips the status of all the bulbs with odd labels (i.e., 1, 3, ...).
  • Button 4: Flips the status of all the bulbs with a label j = 3k + 1 where k = 0, 1, 2, ... (i.e., 1, 4, 7, 10, ...).

You must make exactly presses button presses in total. For each press, you may pick any of the four buttons to press.

Given the two integers n and presses, return the number of different possible statuses after performing all presses button presses.

Key Insights

  • The initial state of all bulbs is ON.
  • Each button press can be treated as toggling specific bulbs, resulting in various combinations of ON and OFF states.
  • The total number of unique bulb states can be derived from the different combinations of button presses.
  • The maximum possible unique states depends on the number of bulbs and the number of presses.

Space and Time Complexity

Time Complexity: O(1) - The logic for determining the number of states can be computed with simple arithmetic based on the number of bulbs and presses. Space Complexity: O(1) - No additional space is required beyond a few variables for calculations.

Solution

The approach to solving this problem involves considering the effects of each button press on the bulbs and how many unique configurations can be achieved after a specified number of presses.

  1. Understanding Button Effects: Each button affects a specific subset of bulbs. We can analyze how these buttons interact when pressed in various sequences.
  2. Dynamic Counting of States: Instead of simulating each press, we can mathematically derive the possible states based on the number of bulbs and the number of button presses.
  3. Unique States Calculation: The total unique states can be derived from the number of bulb combinations that can be achieved through the multiple possible button presses.

Code Solutions

def bulbSwitch(n: int, presses: int) -> int:
    if presses == 0:
        return 1  # Only the initial state
    if presses == 1:
        return min(4, n + 1)  # Maximum unique states in one press
    # For more than one press, determine unique states based on bulbs
    if n == 1:
        return 2  # Two states: on/off
    if n == 2:
        return 3  # Three states: [on, off], [off, on], [off, off]
    # For more bulbs and presses, generalize the counting
    return (1 << presses) if n > 3 else (1 << presses) - 1  # 2^presses unique states

# Example usage
print(bulbSwitch(3, 1))  # Output: 4
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