The Coffee-Milk Conundrum: Exploring Invariants in Liquid Transfer

Q)
There are two equally sized cups: cup 1 contains coffee and cup 2 contains milk. Both cups are half full (we are optimists). Your favorite drink is 1/3 coffee and 2/3 milk. Can you get such a drink in cup 1 by transferring (any amount of) liquid between the two cups? Any amount of your favorite drink would work --- the right proportion is what matters.

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A)
It's impossible to create a drink with 1/3 coffee and 2/3 milk in Cup 1 by pouring liquid between two cups, one initially containing only coffee (Cup 1) and the other only milk (Cup 2). It uses the concept of an "invariant," a property that remains true throughout the process.

Here's the breakdown of the meaning:

1. Initial State: At the beginning, Cup 1 contains only coffee, and Cup 2 contains only milk.
2. Pouring from Cup 1 to Cup 2: When we pour from Cup 1 (coffee) into Cup 2 (milk), we're removing only coffee from Cup 1. This doesn't change the fact that Cup 1 either contains only coffee or a mix where at least half of it is coffee (since we started with all coffee).
3. Pouring from Cup 2 to Cup 1: When we pour from Cup 2 (which now contains milk and some coffee) into Cup 1, we're adding more liquid to Cup 1, but this liquid contains at least as much milk as coffee. Since Cup 2 had more milk to begin with and we only added coffee to it, it still has at least as much milk as coffee. Therefore, pouring from Cup 2 to Cup 1 will always result in Cup 1 having at least as much coffee as milk.
4. Conclusion: Because of these pouring dynamics, Cup 1 will always contain at least as much coffee as milk. It's impossible to get the drink in Cup 1 to have more milk than coffee, which is required for the 1/3 coffee and 2/3 milk ratio. Hence, we cannot achieve the desired proportion in Cup 1 by just transferring liquids between the two cups.

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