**π‘ Problem Formulation:** We need a Pythonic solution to determine whether a given integer number has only the first (highest-order) and last (lowest-order) bits set to 1, with all other bits set to 0. For instance, if the input number in binary is `10000001`

, we expect a result that confirms the number matches our criteria, whereas a number like `10100001`

should not.

## Method 1: Bitwise Operations and Masking

This method involves using bitwise operations to construct a mask that isolates the first and last bits. We then compare this mask against the original number to check if only these two bits are set. It provides a clear and computer science-based approach to solving the problem.

Here’s an example:

def has_first_last_bits_set(num): if num == 0: return False last_bit = 1 first_bit = 1 << (num.bit_length() - 1) mask = first_bit | last_bit return num == mask print(has_first_last_bits_set(129)) # Example number: 10000001 in binary

Output:

True

The function `has_first_last_bits_set`

constructs a mask with only the first and last bits set by shifting `1`

to the left by the number of bits in `num`

minus one. It then checks if the original number is equal to this mask, thereby confirming if only the first and last bits are set.

## Method 2: String Manipulation

String manipulation involves converting the number into binary format, ensuring that the first and last characters are ‘1’, and all other characters, if present, are ‘0’. This method leverages Python’s string manipulation prowess to solve the problem succinctly.

Here’s an example:

def has_first_last_bits_set_str(num): binary_str = bin(num)[2:] # Convert to binary and remove '0b' prefix return binary_str.startswith('1') and binary_str.endswith('1') and binary_str.strip('10') == '' print(has_first_last_bits_set_str(129)) # Example number: 10000001 in binary

Output:

True

The function `has_first_last_bits_set_str`

checks if the binary string representation of the number begins and ends with a ‘1’, and that all characters between can be removed by stripping ‘1’s and ‘0’s, thus confirming that no other ‘1’s are present.

## Method 3: Logarithmic Approach

The logarithmic method calculates if the position of the highest-order bit (first bit) corresponds to a number where only that bit and the lowest-order bit (last bit) are set. It efficiently quantifies the bit positions while accounting for the value of the number being checked.

Here’s an example:

import math def has_first_last_bits_set_log(num): if num == 0: return False last_bit_pos = 0 first_bit_pos = math.floor(math.log(num, 2)) return num == (1 << first_bit_pos) | (1 << last_bit_pos) print(has_first_last_bits_set_log(129)) # Example number: 10000001 in binary

Output:

True

The `has_first_last_bits_set_log`

function computes the highest bit’s position using the base-2 logarithm of `num`

. It checks whether the number is equal to a binary number with ones only in the positions found, indicating that only the first and last bits are set.

## Method 4: Iterative Binary Search

This approach involves iteratively comparing and setting the bits from both the ends of the number until they converge. It suits scenarios where bitwise operations may not be as intuitive, offering a more algorithmic solution.

Here’s an example:

def has_first_last_bits_set_iter(num): bit_length = num.bit_length() left, right = 1 < right and num & left == 0: left >>= 1 while left > right and num & right == 0: right <<= 1 return num == left | right print(has_first_last_bits_set_iter(129)) # Example number: 10000001 in binary

Output:

True

The `has_first_last_bits_set_iter`

function uses a while loop to move the left and right comparators toward each other, checking if there are any other bits set apart from the first and last. It concludes by ensuring the original number is equivalent to just the left and right bits being set.

## Bonus One-Liner Method 5: Direct Combination Check

A direct combination of bitwise shifting and logical assertions can be a compact and Pythonic one-liner solution. It can be less intuitive but works perfectly for those familiar with bitwise operations.

Here’s an example:

has_first_last_bits_set_one_liner = lambda num: num & (num - 1) == 0 and num.bit_length() > 1 print(has_first_last_bits_set_one_liner(129)) # Example number: 10000001 in binary

Output:

True

The one-liner lambda function checks that the number and the number minus one have no common set bits, which would be the case if the number had only the first and last bits set. The additional condition ensures there are at least two bits in the number.

## Summary/Discussion

**Method 1: Bitwise Operations and Masking.**Provides a direct solution using fundamental computer science concepts. Its strengths include clarity and precision. However, it may be less intuitive to those not familiar with bitwise operations.**Method 2: String Manipulation.**Employs Pythonβs string handling capabilities to achieve the goal, which is easy to understand and read. Its downside could be performance on very large numbers due to conversion to strings.**Method 3: Logarithmic Approach.**Efficiently determines the presence of the first and last bits with mathematical functions. It is elegant but requires an understanding of logarithms, which might not be straightforward for all programmers.**Method 4: Iterative Binary Search.**Provides an algorithmic approach suitable for teaching binary bit manipulation. The iterative nature makes it somewhat slower than other methods.**Bonus Method 5: Direct Combination Check.**Offers a succinct and effective one-liner solution. Itβs compact but may sacrifice readability for brevity.

Emily Rosemary Collins is a tech enthusiast with a strong background in computer science, always staying up-to-date with the latest trends and innovations. Apart from her love for technology, Emily enjoys exploring the great outdoors, participating in local community events, and dedicating her free time to painting and photography. Her interests and passion for personal growth make her an engaging conversationalist and a reliable source of knowledge in the ever-evolving world of technology.