Buy Candy - Python Solution and Time Complexity Analysis

A box of candy costs price dollars. You have balance dollars. Compute the number of boxes of candy you can buy and return how many more dollars you need to buy one more box of candy.

Example:

Input: price = 4, balance = 9

Output: 3

Explanation:
You can buy two boxes of candy that costs 4 * 2 = 8 dollars.
If you would have 12 dollars you could buy 3 boxes (one more), so you would need 3 more dollars.

Understanding the Problem

The core challenge of this problem is to determine how many boxes of candy you can buy with a given amount of money and how much more money you would need to buy one additional box. This problem is significant in scenarios where budgeting and incremental purchasing decisions are involved.

Potential pitfalls include miscalculating the number of boxes you can buy or the additional amount needed due to incorrect arithmetic operations.

Approach

To solve this problem, we can break it down into the following steps:

1. Calculate the number of boxes you can buy with the given balance.
2. Determine the total cost if you were to buy one more box.
3. Compute the difference between the required amount for one more box and the current balance.

Let's start with a naive approach and then optimize it.

Naive Approach

The naive approach involves directly calculating the number of boxes and the additional amount needed. This approach is straightforward but can be optimized for clarity and efficiency.

Optimized Approach

The optimized approach involves using integer division and multiplication to determine the number of boxes and the additional amount needed. This approach is more efficient and easier to understand.

Algorithm

Here is a step-by-step breakdown of the optimized algorithm:

1. Calculate the number of boxes you can buy using integer division: boxes = balance // price.
2. Calculate the total cost for one more box: balanceNeeded = (boxes + 1) * price.
3. Compute the additional amount needed: additionalAmount = balanceNeeded - balance.

Code Implementation

def buy_candy(price, balance):
    # Calculate the number of boxes we can buy
    boxes = balance // price
    
    # Calculate the total cost if we buy one more box
    balance_needed = (boxes + 1) * price
    
    # Calculate the additional amount needed to buy one more box
    additional_amount = balance_needed - balance
    
    return additional_amount

# Example usage
price = 4
balance = 9
print(buy_candy(price, balance))  # Output: 3

Complexity Analysis

The time complexity of this solution is O(1) because it involves a constant number of arithmetic operations. The space complexity is also O(1) as we are using a fixed amount of additional space for variables.

Edge Cases

Potential edge cases include:

• balance is less than price: The function should correctly return the difference between price and balance.
• balance is exactly a multiple of price: The function should return price as the additional amount needed.

Example edge cases:

print(buy_candy(4, 3))  # Output: 1
print(buy_candy(4, 8))  # Output: 4

Testing

To test the solution comprehensively, consider a variety of test cases:

• Simple cases where balance is less than, equal to, or greater than price.
• Edge cases where balance is zero or a multiple of price.

Example test cases:

assert buy_candy(4, 9) == 3
assert buy_candy(4, 3) == 1
assert buy_candy(4, 8) == 4
assert buy_candy(4, 0) == 4

Thinking and Problem-Solving Tips

When approaching such problems, consider breaking down the problem into smaller, manageable steps. Use integer division and modular arithmetic to simplify calculations. Practice similar problems to improve your problem-solving skills.

Conclusion

In this blog post, we discussed how to solve the problem of determining the number of candy boxes you can buy and the additional amount needed for one more box. We explored a step-by-step approach, provided a Python implementation, and analyzed the complexity. Understanding and solving such problems is crucial for developing strong problem-solving skills.

Additional Resources

For further reading and practice, consider the following resources:

• LeetCode - Practice coding problems.
• GeeksforGeeks - Tutorials and problem-solving tips.
• Python Official Documentation - Learn more about Python.