Bulb Switcher II

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

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;
    }
}