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There is a room with n bulbs, numbered from 1 to n, arranged in a row from left to right. Initially, all the bulbs are turned off. You have a remote control that can perform exactly m operations, each of which is one of the following:

  1. Flip all the bulbs (i.e., turn each bulb that is off to on and each bulb that is on to off).
  2. Flip all the bulbs with even numbers (i.e., bulbs 2, 4, 6, …).
  3. Flip all the bulbs with odd numbers (i.e., bulbs 1, 3, 5, …).
  4. Flip all the bulbs with numbers that are 3k + 1 (i.e., bulbs 1, 4, 7, …).

Return the number of different possible states of the bulbs after performing m operations.

Example 1:

Input: n = 1, m = 1
Output: 2
Explanation: 
Initially, there is only one bulb. After performing one operation, there are only two possible states: turn on or off.

Example 2:

Input: n = 2, m = 1
Output: 3
Explanation: 
Initially, there are two bulbs. After one operation, three possible states are: [on, off], [off, on], and [on, on].

Example 3:

Input: n = 3, m = 1
Output: 4

Clarifying Questions

  1. What is the maximum value of n and m?
  2. Do all operations need to be different, or can the same operation be performed multiple times?
  3. Are there guaranteed constraints like 1 <= n <= 1000 and 0 <= m <= 1000?

Strategy

The key insight here is to recognize the patterns formed by the operations. For instance:

For smaller values of n and m, the problem can essentially be enumerated and the results pre-determined based on combinations of these operations. Given constraints typically found in interview settings, we can summarize this:

Simplified Problem Analysis by Cases:

  1. When n <= 2: The number of bulbs is very small, making enumeration of state combinations fairly easy.
  2. When m = 0: No operations lead to only one possible state.
  3. When m = 1: Analyzing the first few operations typically spans from complete OFF to combinations of flips.
  4. When m > 1: Combinations of operations will exponentially increase the number of states.

Approach:

We’ll use a set to record possible states and then perform the operations sequentially, considering special scenarios for n (like n=1 or n=2).

Here’s the Python code to implement this:

def flipLights(n: int, m: int) -> int:
    # When no operation is performed
    if m == 0:
        return 1
    
    # When the number of bulbs is 1
    if n == 1:
        return 2 if m > 0 else 1
    
    # When the number of bulbs is 2
    if n == 2:
        if m == 1:
            return 3
        else:
            return 4
    
    # For general case when n >= 3
    if m == 1:
        return 4
    elif m == 2:
        return 7
    else:
        return 8

# Test with example cases
print(flipLights(1, 1)) # Expected output: 2
print(flipLights(2, 1)) # Expected output: 3
print(flipLights(3, 1)) # Expected output: 4

Explanation:

  1. For n = 1, we have 2 states: [on] or [off].
  2. For n = 2 and m = 1, we can achieve 3 distinct configurations.
  3. For larger values of m and n = 2, 4 configurations emerge.
  4. For n >= 3, based on m, configurations grow like 4, 7, or 8 (placed within constraints due to operation combinatory limits).

Time Complexity

The time complexity is O(1) due to the constrained size of n and predictable number of operation combinations.

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