Question:

Here’s a glass of water. The water is in a transparent glass that is a perfect right cylinder. It appears that the glass is half full, but how can you really be sure? How can you accurately determine whether the glass is half full, more than half full, or less than half full? You have no rulers, writing utensils, or other tools.

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Solution: Let’s start off with a solution that most candidates think of first and examine why it’s not good enough. Holding the glass upright, the candidate uses the palm of her left hand to cover the glass. Now she makes a pinching gesture with the index finger and thumb of her right hand. She puts her thumb at the base of the glass and her index finger adjacent to the water level, thereby gauging the height of the water surface from the base of the glass. Now she freezes the distance between her two fingers. She then flips the glass upside down with her left hand; no water falls out since she’s sealed the opening with her left palm. Next she puts her frozen right hand, acting as a gauge, against the glass and checks to see if the inverted water level aligns with her index finger. If so, the glass is exactly half full. This may seem like a good solution, but it’s actually inaccurate, because the palm of the hand is not a perfectly flat surface. Also, the technique will most likely lose some water in the inversion.

The most elegant solution requires the insight that the geometry of the glass offers an absolutely precise solution. The solution is easier to demonstrate than to describe. Carefully tilt the glass toward you so that the water almost spills out, but doesn’t. The geometry of the glass (remember, it’s a perfect right cylinder) is such that if the glass is exactly half full, the water level in the inside of the glass will be touching the upper inside rim of the bottom. If any of the inside bot-tom surface of the glass is exposed, the glass is less than half full. If the level is above the inside bottom ring of the glass, it is more than half full.

Answer: Exploit the inherent geometry of the glass by noting that a perfect right cylinder is half full at the point when the liquid level simultaneously touches the outside rim and inside rim of the glass.