The “100 Coins for 100 Chickens” puzzle, also known as the “Hundred Chickens for a Hundred Coins” problem, is a classic mathematical and algorithmic challenge. It involves finding all possible combinations of three types of chickens (roosters, hens, and chicks) that can be purchased with exactly 100 coins, given their specific costs, and ensuring that the total number of chickens equals 100. In this blog post, we’ll delve into the intricacies of writing a Python program to solve this puzzle.
Understanding the Puzzle
The puzzle’s constraints are as follows:
- 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 find all valid combinations of these chickens that add up to both 100 in terms of the total number of chickens and 100 in terms of the total cost.
Algorithm Design
To solve this puzzle, we’ll use a brute-force approach, iterating through all possible combinations of roosters, hens, and chicks. However, to optimize the search, we’ll apply bounds to the iteration ranges based on the costs and the total number of chickens.
Python Implementation
Here’s a Python program that implements the solution:
python# Initialize an empty list to store valid solutions
solutions = []
# Iterate through possible numbers of roosters (up to 20 since 5*21 > 100)
for roosters in range(21):
# Iterate through possible numbers of hens (up to 33 since 3*34 + 5*20 > 100)
for hens in range(34):
# Calculate the cost of roosters and hens
cost_spent = 5 * roosters + 3 * hens
# Calculate the remaining coins for chicks
remaining_coins = 100 - cost_spent
# Calculate the number of chick groups (each group costs 1 coin)
chick_groups = remaining_coins
# Check if the number of chick groups is non-negative and the total number of chickens is 100
if chick_groups >= 0 and roosters + hens + chick_groups * 3 == 100:
# Convert chick groups to total number of chicks
chicks = chick_groups * 3
# Add the 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]}")
Efficiency and Optimization
While the brute-force approach works well for this problem due to its constrained nature, it might not be suitable for more complex or larger-scale problems. For those, more advanced algorithms and optimization techniques would be necessary. However, for this specific puzzle, the implemented solution efficiently finds all valid combinations within a reasonable time frame.
Conclusion
The “100 Coins for 100 Chickens” puzzle is a great exercise in algorithmic thinking and problem-solving. By writing a Python program to solve it, we’ve not only developed our coding skills but also honed our ability to approach and solve mathematical and logical challenges. This program serves as a solid foundation for tackling more complex algorithmic problems in the future.
