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Bulb Switcher II

Updated on 14 May, 2025
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Problem Statement

Imagine a room equipped with n light bulbs, each individually labeled from 1 to n. When you first enter, all bulbs are turned on. On the wall, you find four distinct buttons, each with its own unique functionality affecting the bulbs' status:

  • Button 1: Toggles the on/off status of every bulb in the room.
  • Button 2: Toggles the on/off status of bulbs that are labeled with an even number (e.g., 2, 4, ...).
  • Button 3: Toggles the on/off status of bulbs that are labeled with an odd number (e.g., 1, 3, ...).
  • Button 4: Toggles the on/off status of bulbs labeled according to the formula j = 3k + 1 for k = 0, 1, 2, ... (such as bulbs 1, 4, 7, 10, ...).

Your challenge is to utilize exactly presses button presses, during which you can press any of the four buttons in any sequence. The goal is to determine how many distinct bulb statuses or configurations can be achieved after all presses have been made.

Examples

Example 1

Input:

n = 1, presses = 1

Output:

2

Explanation:

Status can be:
- [off] by pressing button 1
- [on] by pressing button 2

Example 2

Input:

n = 2, presses = 1

Output:

3

Explanation:

Status can be:
- [off, off] by pressing button 1
- [on, off] by pressing button 2
- [off, on] by pressing button 3

Example 3

Input:

n = 3, presses = 1

Output:

4

Explanation:

Status can be:
- [off, off, off] by pressing button 1
- [off, on, off] by pressing button 2
- [on, off, on] by pressing button 3
- [off, on, on] by pressing button 4

Constraints

  • 1 <= n <= 1000
  • 0 <= presses <= 1000

Approach and Intuition

When tackling this problem, the primary consideration is how each button affects the bulbs and how these effects can overlap or nullify each other. Here's a breakdown of the approach and the underlying thought process:

  1. Identify Bulb Groups:

    • Each button affects a distinct group of bulbs, based on either numerical order or a mathematical formula.
    • Understand how these groups intersect (e.g., bulbs affected by both Button 1 and Button 2, or Button 2 and Button 3), as overlapping effects can lead back to the original state or toggle a specific subset.

  2. Modeling Button Effects:

    • Recognize that pressing a button twice reverses its effect, rendering it as if it was never pressed. This means that pressing any button an even number of times yields no effect on the bulb's initial status.
    • Identify unique state changes. For instance, pressing Button 1, then Button 2, may not be the same as pressing Button 2 followed by Button 1 due to the distribution and toggle pattern of the bulbs.

  3. Dynamic Configurations:

    • With presses operations and four buttons, explore the combinations of presses - remember that multiple presses of the same button can ultimately negate each other.
    • Consider the effect of combinations on bulb states – some combinations will achieve the same end state, reducing the total count of unique configurations.

  4. Tracking State Modifications:

    • Use a methodical approach to simulate each possible sequence of button presses and document the resulting bulb configurations.
    • Implementing a computational model would involve creating a matrix or array to simulate bulb states and applying button effects sequentially to see the end result.

The nuances of button interactions and their collective impacts on the system state make this a challenging combinatorial problem, one suited for algorithmic simulation to enumerate and deduce the distinct configurations possible, considering all the rules and constraints.

Solutions

  • Java
  • Python
java 
class Solution {
    public int toggleLights(int lights, int presses) {
        lights = Math.min(lights, 3);
        if (presses == 0) return 1;
        if (presses == 1) return lights == 1 ? 2 : lights == 2 ? 3 : 4;
        if (presses == 2) return lights == 1 ? 2 : lights == 2 ? 4 : 7;
        return lights == 1 ? 2 : lights == 2 ? 4 : 8;
    }
}

In the "Bulb Switcher II" problem, the solution in Java focuses on toggling a set number of lights with a given number of presses. This solution takes advantage of the repeating pattern that emerges due to the limited interactions between toggles. Here's a brief explanation of the implemented logic:

  • The number of lights considered is limited to a maximum of three. This simplifies the problem since beyond three lights, patterns start repeating due to the symmetric nature of toggle operations.

  • The method checks for:

    • Zero Presses: Always returns 1 since no lights change their state.
    • One Press: The number of possible configurations depends directly on the number of lights.
      • For one light, can toggle on or off, thus 2 configurations.
      • For two lights, can result in 3 different states.
      • For three lights, can result in 4 different states.
    • Two Presses: As before, results depend on the number of lights:
      • For one light, results in 2 configurations as it can either stay on or be off.
      • For two lights, 4 possible outcomes are observed.
      • For three lights, 7 different configurations are possible.
    • Three or More Presses: Patterns stabilize beyond three presses, resulting in set outcomes:
      • For one light, the result is similar to having two presses.
      • For two lights, the results do not change.
      • For three lights, there are a constant eight configurations.

This solution is efficient owing to its reduced focus beyond three lights and presses, keying into the repetitive nature of possible light configurations with each press. Notice how the solution appropriately uses the minimum function to prevent unnecessary calculations and conditionals to manage different outcomes based on the counts of lights and presses.

python 
class Solution(object):
    def toggleLights(self, count, moves):
        count = min(count, 3)
        if moves == 0: return 1
        if moves == 1: return [2, 3, 4][count-1]
        if moves == 2: return [2, 4, 7][count-1]
        return [2, 4, 8][count-1]

This Python solution addresses the "Bulb Switcher II" problem. It efficiently calculates the number of different configurations possible for up to three light bulbs, after any given number of moves. The core logic limits the maximum number of bulbs to consider to three, leveraging the fact that all possible configurations for larger sets can be reduced to this size.

Understand the toggleLights function:

  • It starts by minimizing the count of bulbs to three to handle scenarios equitably regardless of the actual number of bulbs.
  • It then evaluates the number of possible configurations based on the value of moves:
    • If moves is 0, it directly returns 1 since no moves mean no change in configuration.
    • For 1 move, the configurations depend on the count of bulbs: 2 configurations for 1 bulb, 3 for 2 bulbs, and 4 for 3 bulbs.
    • For 2 moves, it returns 2 configurations for 1 bulb, 4 for 2 bulbs, and 7 for 3 bulbs.
    • For more than 2 moves, the pattern only slightly varies from the 2-moves scenario.

The solution uses a straightforward conditional structure and list indexing to yield results. This approach ensures optimal performance for the restricted problem scenario and provides instant results by simplifying the decision-making down to indexed list access.

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