The Experiment
Wine Glass
Water Glass
The Puzzle
The Question
You have a glass of wine and a glass of water, filled equally.
- Take a spoonful of wine and add it to the water
- Stir the water glass thoroughly
- Take a spoonful of the mixture and add it back to the wine
Is there more wine in the water glass, or more water in the wine glass?
What Most People Think
The first spoonful is pure wine. The second spoonful is diluted. So surely there's more wine in the water than water in the wine?
This intuition is wrong!
The Answer: Exactly Equal!
No matter how much you transfer or how well you stir, the amount of wine in the water glass always equals the amount of water in the wine glass!
Why It Works
The Conservation Argument
At the end, both glasses have the same total volume as they started with.
If some wine is "missing" from the wine glass, it must be in the water glass.
But the water glass is back to its original volume. So whatever wine is there has displaced an equal volume of water.
That displaced water? It's now in the wine glass!
The Marble Proof
Imagine 50 red marbles (wine) and 50 blue marbles (water).
- Move 10 red marbles to the blue pile (now 40 red, 60 mixed)
- Move 10 marbles back (some red, some blue)
- Each pile has 50 marbles again
If x blue marbles are now in the red pile, then x red marbles must be in the blue pile. It's just conservation!
The Deep Insight
It's Even More General
This result holds no matter what:
- The size of the spoon (as long as it's the same both times)
- How well you stir (or don't stir at all!)
- How many transfers you make (as long as glasses end at original levels)
- The shapes of the glasses
The Key Constraint
The only requirement is that both glasses end up at their original volumes. This conservation constraint forces the "contaminations" to be equal.
Historical Notes
This puzzle appears in Martin Gardner's "Hexaflexagons and Other Mathematical Diversions" (1959) and was also discussed by George Gamow and Marvin Stern in "Puzzle Math" (1958). It's been stumping people for decades!