7: Egg-Drop Soup
To celebrate the end of classes (and our final puzzle of the year) the physics department is planning to make their famous egg drop soup. To make egg drop soup, you have to break eleven eggs by dropping them from the lowest step between the first and second floors of the Science Center which will cause them to break. If you drop an egg from too low a step, the egg will not break. If you drop an egg from too high a step, the soup will be ruined. There are twenty-one steps.
Here are some more guidelines: If an egg does not break, it can be re-used. If one egg breaks, or fails to break, from a given step, then all the other eggs will do the same. If an egg breaks when dropped from a given step, then all eggs will break when dropped from all higher steps. If an egg does not break when dropped from a given step, then all eggs will not break when dropped from all lower steps.
The physicists have a dozen eggs. If they needed the whole dozen for the soup, they would have to start at the first step, dropping an egg at each step until it broke. Once they reached the lowest step at which the egg breaks, they would drop the rest of the dozen, scrape them from the floor, and make the soup. This procedure might require as many as 21 egg drops.
But, they only need eleven eggs.
What is the fewest number of drops that is guaranteed to determine the right height from which to drop eggs, and which will lead to yummy egg-drop soup? Solutions must include an explanation.
Can you generalize the result to n steps?