**π‘ Problem Formulation:** In binary sequences, finding the value of a specific bit at position `k`

within the `n`

th binary number is a common task. Given an integer `n`

, consider this as a binary string after converting to binary format. The challenge is to determine the value of the bit at the `k`

th position (from the right) in this binary string. For instance, if `n`

is 9 (binary ‘1001’) and `k`

is 2, the desired output is ‘0’, which is the second bit from the right.

## Method 1: Using Bitwise Operations

Performing a bitwise ‘AND’ operation between the number `n`

shifted to the right by `k-1`

places and the number 1 yields the `k`

th bit. This is because shifting brings the kth bit to the least significant bit position, and a bitwise ‘AND’ with 1 will isolate this bit.

Here’s an example:

def find_kth_bit(n, k): return (n >> (k-1)) & 1 print(find_kth_bit(9, 2)) # Example with n=9 and k=2

Output: 0

This code snippet defines a simple function `find_kth_bit()`

that uses right bit shifting and a bitwise ‘AND’ to retrieve the kth bit of the binary representation of n. It shifts `n`

to the right by `k-1`

and then applies an ‘AND’ with 1 to extract the least significant bit, which is our kth bit.

## Method 2: Using Binary String Conversion

This method involves converting the number `n`

to its binary string representation and then accessing the `k`

th bit from the right directly by indexing the string. Negative indices can be used to access bits from the end of the string.

Here’s an example:

def find_kth_bit_string(n, k): return bin(n)[-k] print(find_kth_bit_string(9, 2)) # Example with n=9 and k=2

Output: ‘0’

The function `find_kth_bit_string()`

first converts `n`

to a binary string using the built-in `bin()`

function. The kth bit from the end is then accessed directly using negative indexing. Note that the binary string contains a ‘0b’ prefix, which is removed by using negative indexing.

## Method 3: Using Iteration and Shifting

To find the kth bit, this method uses a loop to shift through each bit one by one until it reaches the kth bit. At each iteration, the least significant bit is checked and the result is updated after each right shift operation.

Here’s an example:

def find_kth_bit_iterative(n, k): for i in range(k-1): n >>= 1 return n & 1 print(find_kth_bit_iterative(9, 2)) # Example with n=9 and k=2

Output: 0

The `find_kth_bit_iterative()`

function loops exactly `k-1`

times, each time shifting the number `n`

one bit to the right. After the loop completes, it performs a bitwise ‘AND’ with 1 to isolate the kth bit. This method mimics the manual process of checking each bit sequentially.

## Method 4: Using Masking

A mask is created with a ‘1’ at the kth bit and ‘0’s elsewhere. Then a bitwise ‘AND’ operation with `n`

will reveal the kth bit. This mask is obtained by left shifting ‘1’ by `k-1`

places.

Here’s an example:

def find_kth_bit_masking(n, k): mask = 1 <> (k-1) print(find_kth_bit_masking(9, 2)) # Example with n=9 and k=2

Output: 0

The `find_kth_bit_masking()`

function creates a mask with a bit pattern that has a ‘1’ only at the kth position when counted from the right. Applying this mask with an ‘AND’ operation to `n`

leaves only the kth bit, which is then shifted back `k-1`

times to the least significant bit position for easy reading.

## Bonus One-Liner Method 5: Using int.__getitem__()

Python’s `int`

objects can be treated like sequences of bits using magic method `__getitem__()`

. To find the kth bit, we simply use this method with the appropriate index.

Here’s an example:

n = 9 k = 2 print((n).__getitem__(k-1)) # Example with n=9 and k=2

Output: 0

This one-liner approach directly accesses the kth bit using the `__getitem__()`

method of integer objects. Negative indices are used to access the sequence from the end, with the rightmost bit being at index `-1`

.

## Summary/Discussion

**Method 1:**Bitwise Operations. Strengths: It’s very efficient and does not require conversion to strings. Weaknesses: Bitwise operations can be less intuitive for those not familiar with them.**Method 2:**Binary String Conversion. Strengths: Straightforward and easy to understand. Weaknesses: It is less efficient due to the string conversion and handling.**Method 3:**Iteration and Shifting. Strengths: Demonstrates a clear step-by-step process. Weaknesses: Inefficient for large values of k.**Method 4:**Masking. Strengths: Conceptually clear and efficient. Weaknesses: Requires additional bitwise understanding.**Method 5:**int.__getitem__(). Strengths: Concise and leverages built-in Python features. Weaknesses: Uses an internal method, which can be considered cryptic or non-idiomatic.