Exploring the “Hundred Money for Hundred Chickens” Problem with Python

The “Hundred Money for Hundred Chickens” problem, a timeless mathematical puzzle, challenges us to find all possible combinations of three types of chickens—roosters, hens, and chicks—that can be purchased with exactly 100 coins, and where the total number of chickens equals 100. In this blog post, we’ll delve into the specifics of using Python programming to implement a solution to this problem, examining the approach, code implementation, and its educational significance.

Problem Statement

The puzzle constraints are straightforward:

• Roosters cost 5 coins each.
• Hens cost 3 coins each.
• Chicks cost 1/3 of a coin each, meaning 3 chicks can be bought for 1 coin.

We need to determine all combinations (r, h, c) where r is the number of roosters, h is the number of hens, and c is the number of chicks, such that r + h + c = 100 and 5r + 3h + c/3 = 100.

Python Implementation

To solve this problem with Python, we’ll employ a nested loop structure to iterate through all possible combinations of roosters and hens, while ensuring that the constraints are met. Here’s a straightforward implementation:

python# Initialize an empty list to store valid solutions
solutions = []

# Iterate through possible numbers of roosters (up to 20, as 5*21 > 100)
for roosters in range(21):
    # Iterate through possible numbers of hens (adjust bounds as necessary)
    for hens in range(34):  # 3*34 + 5*20 = 102, close to 100 for efficiency
        # Calculate the number of chicks based on the remaining money and total chickens
        chicks = 100 - roosters - hens
        # Check if the combination satisfies the cost constraint
        if (5 * roosters + 3 * hens + chicks // 3) == 100 and chicks % 3 == 0:
            # Append the valid solution to the list
            solutions.append((roosters, hens, chicks))

# Print the solutions
for solution in solutions:
    print(f"Roosters: {solution[0]}, Hens: {solution[1]}, Chicks: {solution[2]}")


Note: The condition chicks % 3 == 0 ensures that the number of chicks is divisible by 3, as each chick costs 1/3 of a coin.
Analyzing the Code

Efficiency: The code efficiently explores the search space by iterating through reasonable bounds for roosters and hens, reducing unnecessary computations.
Logic: The logic is straightforward, utilizing arithmetic operations and conditional statements to enforce the constraints.
Output: The program prints all valid solutions, making the results easy to interpret.

Educational Value

Problem-solving: Encourages critical thinking and algorithmic problem-solving skills.
Python Fundamentals: Reinforces Python basics like loops, conditional statements, and arithmetic operations.
Mathematical Modeling: Illustrates how mathematical constraints can be modeled and solved programmatically.
Debugging and Testing: Provides an opportunity to practice debugging and testing, as the initial implementation might need adjustments to perfectly align with the problem’s constraints.

Conclusion
The “Hundred Money for Hundred Chickens” problem, solved with Python, offers a rich learning experience. It combines algorithmic thinking, mathematical modeling, and programming skills, making it an excellent exercise for students and enthusiasts alike. Through this journey, we’ve not only found the solutions to the puzzle but also strengthened our understanding of programming and problem-solving techniques.