The “Hundred Chickens for a Hundred Coins” problem is a classic mathematical puzzle that challenges us to find all possible combinations of chickens bought with a hundred coins, under certain constraints. Specifically, a farmer has a hundred coins and wants to buy exactly a hundred chickens. The chickens come in three types:
- Roosters cost 5 coins each.
- Hens cost 3 coins each.
- Chicks cost 1/3 of a coin each (or, equivalently, 3 chicks cost 1 coin).
In this blog post, we’ll explore how to solve this problem using Python, breaking down the solution into clear, manageable steps.
Step 1: Understanding the Problem
Before diving into the code, it’s essential to clearly understand the problem’s constraints and objectives. We need to find all integer values for roosters (R), hens (H), and chicks (C) that satisfy the following conditions:
- R + H + C = 100 (total number of chickens)
- 5R + 3H + C/3 = 100 (total cost in coins, where C/3 represents the cost of chicks)
Note that since chicks come in groups of 3 for 1 coin, we can replace C with 3c, where c is the number of groups of 3 chicks. This simplifies the second equation to:
- 5R + 3H + c = 100
Step 2: Formulating the Solution
With the constraints in place, we can now formulate a solution using Python. We’ll use nested loops to iterate through possible values of R, H, and c (or, equivalently, C = 3c), checking if they satisfy the constraints.
Step 3: Implementing the Solution
Here’s how the solution might look in Python:
python# Initialize an empty list to store valid solutions
solutions = []
# Iterate through possible values of roosters (R)
for R in range(21): # Maximum 20 roosters since 5*21 > 100
# Iterate through possible values of hens (H)
for H in range(34): # Maximum 33 hens since 3*33 + 5*20 > 100
# Calculate the number of groups of 3 chicks (c)
c = 100 - 5*R - 3*H
# Check if c is an integer (since chicks must come in groups of 3)
if c % 1 == 0:
# Convert c groups of 3 chicks to total number of chicks (C)
C = 3 * c
# Check if the total number of chickens is 100
if R + H + C == 100:
# Add the solution to the list
solutions.append((R, H, C))
# Print the solutions
for solution in solutions:
print(f"Roosters: {solution[0]}, Hens: {solution[1]}, Chicks: {solution[2]}")
Step 4: Reviewing the Results
Running the code above will produce all valid combinations of roosters, hens, and chicks that satisfy the problem’s constraints. Each combination represents a possible way for the farmer to buy exactly a hundred chickens with a hundred coins.
Step 5: Analyzing the Solution
The solution presented here is straightforward and easy to understand, thanks to the use of nested loops and basic arithmetic operations. However, it’s worth noting that the efficiency of this approach can be improved by reducing the range of values that the loops iterate over, based on the constraints of the problem.
Conclusion
The “Hundred Chickens for a Hundred Coins” problem provides an excellent opportunity to practice problem-solving skills and apply basic programming concepts in Python. By breaking down the problem into manageable steps and carefully formulating the solution, we can efficiently find all valid combinations of chickens that meet the given constraints.
