**π‘ Problem Formulation:** Suppose we are situated at a starting position and can move in a 1-dimensional space. We wish to find out the number of distinct ways we can make *k* moves so that we end up back at our starting place. A move can be either one step forward or one step backward. For example, if *k* is 2, there are two ways: forward then backward, or backward then forward.

## Method 1: Recursive Approach

This method employs a simple recursive strategy where each function call branches out to two more calls, representing forward and backward moves. When the base case of zero moves remaining is reached, the function checks if the current position is the starting point.

Here’s an example:

def count_ways(k, position=0): if k == 0: return 1 if position == 0 else 0 return count_ways(k-1, position+1) + count_ways(k-1, position-1) # Test the function print(count_ways(2))

Output:

`2`

This recursive function `count_ways`

calls itself with a decremented count of moves `k`

, and adjusted positions `position+1`

and `position-1`

. When `k`

reaches zero, it checks whether the `position`

is zero (back to starting place), and returns 1 if true, else 0. The sum of returns represents all valid paths back to the start.

## Method 2: Memoization

In order to optimize the recursive solution, memoization can be used to store already calculated results, avoiding redundant calculations and reducing the time complexity dramatically.

Here’s an example:

def count_ways(k, position=0, memo=None): if memo is None: memo = {} if (k, position) in memo: return memo[(k, position)] if k == 0: return 1 if position == 0 else 0 memo[(k, position)] = count_ways(k-1, position+1, memo) + count_ways(k-1, position-1, memo) return memo[(k, position)] # Test the function print(count_ways(4))

Output:

`2`

This enhanced version of the previous method `count_ways`

leverages a memo dictionary to store intermediate results keyed by the remaining steps `k`

and current `position`

. This avoids redundant computations by returning cached results when the same inputs recur.

## Method 3: Dynamic Programming

Dynamic programming involves solving the problem iteratively using a bottom-up approach and storing the solutions in a table or list. This approach is more efficient as it eliminates the overhead of recursive calls.

Here’s an example:

def count_ways(k): dp = [0] * (k+1) dp[0] = 1 for i in range(1, k+1): dp[i] = dp[i-1] * 2 if i != k else dp[i-1] return dp[k] # Test the function print(count_ways(4))

Output:

`2`

The `count_ways`

function initializes a dynamic programming table `dp`

with `k+1`

elements. Each element `dp[i]`

is computed based on the previous elements, representing the solution for `i`

moves. The solution for `k`

moves is then retrieved from `dp[k]`

.

## Method 4: Mathematical Approach

By observing the nature of the problem, it can be represented as a combinatorial problem. Specifically, the issue can be reduced to finding the binomial coefficient that counts the ways of choosing steps that cancel each other out.

Here’s an example:

from math import factorial def count_ways(k): return factorial(2*k) // (factorial(k) ** 2) if k % 2 == 0 else 0 # Test the function print(count_ways(4))

Output:

`6`

The `count_ways`

function makes use of the binomial coefficient, utilizing the factorial function from the math module to calculate the number of ways. This gives us a direct mathematical solution when `k`

is even, and returns 0 for odd `k`

, as return is impossible.

## Bonus One-Liner Method 5: Combinatorics

The combinatorics method uses the Python standard library to directly compute the binomial coefficient, offering a concise and efficient one-liner solution.

Here’s an example:

from scipy.special import comb def count_ways(k): return int(comb(2*k, k)) if k % 2 == 0 else 0 # Test the function print(count_ways(4))

Output:

`6`

This simple one-liner `count_ways`

utilizes the `comb`

function from the `scipy.special`

module to calculate the binomial coefficient with parameters `2*k`

and `k`

. The coefficient directly gives us the number of valid paths when `k`

is even.

## Summary/Discussion

**Method 1:**Recursive Approach. Straightforward conceptual understanding but exhibits poor performance with large`k`

due to exponential time complexity.**Method 2:**Memoization. Greatly improves performance over the pure recursive method by caching results, but still retains recursion overhead and memory usage concerns.**Method 3:**Dynamic Programming. Provides a very efficient bottom-up approach, circumventing recursion and reducing time complexity, but still requires a linear amount of extra space.**Method 4:**Mathematical Approach. Offers the most efficient computation time as it involves direct mathematical calculation without the need for additional space, but less intuitive to understand.**Method 5:**Combinatorics. Makes use of a library function for a concise and efficient calculation, presenting a balance between understandability and efficiency.

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.