The “Hundred Chickens for a Hundred Coins” problem is a timeless algorithmic challenge that serves as an excellent introduction to problem-solving and computational thinking. In this blog post, we’ll delve into the specifics of the algorithm used to solve this problem in Python, discussing its underlying logic, efficiency, and practical implications.
Problem Definition
The problem requires finding all possible combinations of three types of chickens (roosters, hens, and chicks) that can be purchased with exactly 100 coins, given their respective costs:
- Roosters cost 5 coins each.
- Hens cost 3 coins each.
- Chicks cost 1/3 of a coin each (effectively, 3 chicks cost 1 coin).
The total number of chickens must also be exactly 100.
Algorithm Overview
The algorithm to solve this problem involves iterating through potential combinations of roosters, hens, and chicks, and checking if they satisfy the constraints. However, to optimize the search, we can employ a few strategies:
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Bounding the Search Space: We can limit the range of possible values for roosters and hens based on their costs. For example, since roosters cost 5 coins each, we can’t buy more than 20 (since 5*21 > 100). Similarly, we can bound the number of hens.
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Simplifying Chick Calculations: Instead of iterating through individual chicks, we can iterate through the number of groups of 3 chicks, since each group costs 1 coin. This significantly reduces the complexity of the search.
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Conditional Checks: For each combination of roosters and hens, we calculate the number of chick groups that can be bought with the remaining coins. We then check if this number is an integer and if the total number of chickens equals 100.
Python Implementation
Here’s a Python program that implements the above algorithm:
python# Initialize an empty list to store valid solutions
solutions = []
# Iterate through possible numbers of roosters
for roosters in range(21): # 0 to 20 since 5*21 > 100
# Iterate through possible numbers of hens
for hens in range(34): # 0 to 33 since 3*34 + 5*20 > 100
# Calculate the number of chick groups that can be bought with the remaining coins
chick_groups = (100 - (5 * roosters + 3 * hens))
# Check if the number of chick groups is an integer and if the total number of chickens is 100
if chick_groups >= 0 and chick_groups % 1 == 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
The algorithm presented above is efficient for this specific problem due to the bounding of the search space and the simplification of chick calculations. However, for more complex problems, further optimizations might be necessary, such as using more advanced data structures or algorithmic techniques.
Conclusion
The “Hundred Chickens for a Hundred Coins” problem is a great way to practice algorithmic thinking and problem-solving skills. By understanding the underlying logic of the algorithm and implementing it in Python, we can develop a deeper appreciation for computational problem-solving. Moreover, the strategies employed in this algorithm, such as bounding the search space and simplifying calculations, can be applied to a wide range of algorithmic challenges.
