Implementing the Monkey and Stairs Problem in Python

The “Monkey and Stairs” problem, also known as the Fibonacci Stairs problem, is a classic dynamic programming puzzle. The scenario is that a monkey is climbing a staircase and can jump either one or two steps at a time. The question is: in how many ways can the monkey climb a staircase of a given number of steps?

Understanding the Problem

Let’s consider a staircase with n steps. If the monkey is at the nth step, it could have come from either the (n-1)th step by jumping one step or from the (n-2)th step by jumping two steps. This means that the total number of ways to reach the nth step is the sum of the ways to reach the (n-1)th step and the (n-2)th step.

This problem is analogous to the Fibonacci sequence, where each number is the sum of the two preceding numbers. The base cases for this problem are:

• If n = 0, there is one way to reach the top (i.e., not moving from the starting point).
• If n = 1, there is one way to reach the top (i.e., jumping one step).

Implementing the Solution in Python

We can use a recursive approach to solve this problem, but it can be inefficient for larger values of n due to redundant calculations. Instead, we can use a dynamic programming approach to store the results of previous calculations and avoid redundant work.

Here’s a Python implementation of the dynamic programming solution:

pythondef monkey_stairs(n):
    if n == 0 or n == 1:
        return 1
    
    # Initialize a list to store the results of previous calculations
    ways = [0] * (n + 1)
    ways[0] = 1
    ways[1] = 1
    
    # Calculate the number of ways to reach each step
    for i in range(2, n + 1):
        ways[i] = ways[i - 1] + ways[i - 2]
    
    return ways[n]

# Example usage
steps = 5
print(f"The number of ways a monkey can climb {steps} stairs is: {monkey_stairs(steps)}")

This implementation initializes a list ways to store the number of ways to reach each step. It then iterates from 2 to n and calculates the number of ways to reach each step by summing the previous two steps. Finally, it returns the number of ways to reach the nth step.
Conclusion
The “Monkey and Stairs” problem is a great example of how dynamic programming can be used to solve optimization problems efficiently. By storing the results of previous calculations, we can avoid redundant work and significantly improve the performance of our solution. In this article, we discussed the problem, understood its relation to the Fibonacci sequence, and implemented a Python solution using the dynamic programming approach.