Water Bottles II

Start by drinking all full bottles while tracking empty ones. Exchange empty bottles for full ones whenever possible and continue the process until no more exchanges are allowed. This approach guarantees the correct maximum count by simulating each step and applying the exchange logic consistently.

Problem Statement

You have a certain number of full water bottles and a rule to exchange empty bottles for new full ones. Each time you drink a full bottle, it becomes empty, and you can trade a set number of empty bottles for one full bottle.

Given two integers numBottles and numExchange, determine the total number of bottles you can drink. You must simulate the drinking and exchanging process without combining multiple exchanges in one step, ensuring you follow the exact exchange rule at each iteration.

Examples

Example 1

Input: numBottles = 13, numExchange = 6

Output: 15

The table above shows the number of full water bottles, empty water bottles, the value of numExchange, and the number of bottles drunk.

Example 2

Input: numBottles = 10, numExchange = 3

Output: 13

The table above shows the number of full water bottles, empty water bottles, the value of numExchange, and the number of bottles drunk.

Constraints

• 1 <= numBottles <= 100
• 1 <= numExchange <= 100

Solution Approach

Iterative Simulation

Use a loop to drink full bottles and collect empties. After each round, trade every numExchange empty bottles for new full bottles and repeat until no more trades are possible.

Mathematical Shortcut

Recognize that total drinks equal initial bottles plus additional bottles from exchanges. Compute total exchanges by integer division and remainder until empty bottles are insufficient.

Edge Case Handling

Ensure that when empty bottles are fewer than numExchange, no exchange occurs. Avoid attempting multiple exchanges simultaneously, as it violates the problem's simulation pattern.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

Time complexity is O(log(numBottles)) if tracking exchanges iteratively, since each exchange reduces empty bottles. Space complexity is O(1) because only counters for full and empty bottles are maintained.

What Interviewers Usually Probe

• Focus on simulating each exchange step carefully rather than jumping to a formula.
• Check whether edge cases like numBottles < numExchange are handled correctly.
• Watch for infinite loops if empty bottles are not reduced after exchanges.

Common Pitfalls or Variants

Common pitfalls

• Trying to combine multiple exchanges at once instead of following the iterative pattern.
• Forgetting to increment the total drink count after each new full bottle obtained.
• Neglecting the scenario when remaining empty bottles are less than numExchange, causing incorrect loops.

Follow-up variants

• Change numExchange to vary at each step, testing dynamic exchange rates.
• Introduce a limit on total exchanges allowed, requiring careful simulation.
• Start with zero full bottles but some empty bottles to see if exchange logic triggers correctly.

FAQ

What is the main approach for Water Bottles II?

Simulate drinking bottles and exchanging empties step by step, or compute total via repeated integer division for exchanges.

Can multiple exchanges happen in one iteration?

No, each exchange must follow the rule exactly; multiple batch exchanges at once violate the problem's simulation pattern.

What if numBottles is less than numExchange?

You can only drink the existing bottles with no exchange possible, ending the simulation.

Does the problem require tracking empty bottles?

Yes, tracking empties is essential to know when exchanges can occur and calculate the total drinks.

Which pattern is Water Bottles II an example of?

It exemplifies Math plus Simulation, combining iterative exchange calculations with a running total of drinks.