Power of Four

The problem asks you to determine whether a given integer ( n ) is a power of four. You can do this by following a simple approach.

  1. Check if ( n ) is Positive: Since powers of four are positive, return false if ( n ) is less than or equal to zero.
  2. Check if ( n ) is Power of Two: A number that is a power of four must also be a power of two. You can check this by ensuring that ( n ) has exactly one ‘1’ bit in its binary representation.
  3. Check if the ‘1’ Bit is in the Right Position: Since powers of four have the ‘1’ bit in odd positions (counting from right to left and starting from 0), you can check this condition to ensure that ( n ) is a power of four.

Here’s the code:

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class Solution:
    def isPowerOfFour(self, n: int) -> bool:
        if n <= 0:
            return False

        # Check if n is a power of two
        if n & (n - 1) != 0:
            return False

        # Check if the '1' bit is in the correct position
        return n & 0x55555555 != 0

Explanation:

  • The first condition checks if ( n ) is positive.
  • The second condition uses bitwise operations to check if ( n ) is a power of two.
  • The third condition uses a bitmask (0x55555555) to check if the ‘1’ bit is in the correct position. In binary, this mask has ‘1’ bits at all odd positions, allowing us to check if ( n ) is a power of four.

This approach ensures an efficient and correct solution for the given problem.

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class Solution:
    def isPowerOfFour(self, n: int) -> bool:
        return n > 0 and log(n, 4).is_integer()

Problem Classification

This problem falls under the following categories:

  1. Mathematical Problems: The problem involves performing mathematical operations to determine if a number is a power of four.

  2. Number Theory: Specifically, it is related to properties of numbers and their powers.

  3. Logarithms: The simplest solution involves using logarithms to determine if a number is a power of four.

  4. Binary Representation: Another approach involves using the binary representation of the number.

  5. Problem Solving: This is a problem-solving task that involves applying the correct mathematical rules and logic to solve the problem.

  6. Boolean Logic: The final output is a Boolean value (true or false) depending on whether the number is a power of four.

Language Agnostic Coding Drills

  1. Understanding and Checking Conditions: The first step is to understand that a number can only be a power of four if it’s positive. Therefore, the first check in our solution should be to verify that the input number is greater than zero.

    Drill: Write a function that takes an integer as input and returns true if the number is greater than zero, false otherwise.

  2. Working with Logarithms: The concept of logarithms is crucial to solve this problem. If the logarithm to the base 4 of a number is an integer, that number is a power of four.

    Drill: Write a function that takes a positive integer as input and calculates its logarithm to the base 4. The function should return the result.

  3. Checking for Integer Results: If the result of the logarithm operation is an integer, the original number is a power of four.

    Drill: Extend the function from the previous step. After calculating the logarithm, check whether the result is an integer or not. Return True if it’s an integer, False otherwise.

  4. Combine Above Steps: After understanding and coding the above drills, we are now ready to solve the problem. We need to combine these steps into a single solution, checking first if the number is positive and then whether the logarithm of the number to the base 4 is an integer.

In the final solution, we should ensure we correctly handle all potential edge cases, such as the input being zero or negative.

Targeted Drills in Python

  1. Understanding and Checking Conditions
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def is_positive(n):
    return n > 0

# Testing
print(is_positive(4))  # Expected output: True
print(is_positive(-1))  # Expected output: False
  1. Working with Logarithms
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from math import log

def log_base_4(n):
    if n <= 0:
        return None
    return log(n, 4)

# Testing
print(log_base_4(4))  # Expected output: 1.0
print(log_base_4(16))  # Expected output: 2.0
  1. Checking for Integer Results
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from math import log

def is_integer_log_base_4(n):
    log_n = log_base_4(n)
    if log_n is None:
        return False
    return log_n.is_integer()

# Testing
print(is_integer_log_base_4(4))  # Expected output: True
print(is_integer_log_base_4(15))  # Expected output: False
  1. Combine Above Steps
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from math import log

def is_power_of_four(n):
    if not is_positive(n):
        return False
    return is_integer_log_base_4(n)

# Testing
print(is_power_of_four(4))  # Expected output: True
print(is_power_of_four(15))  # Expected output: False

With these drills, you should be able to understand each step involved in the final solution and practice the concepts of checking conditions and working with logarithms.