Wednesday, April 3, 2013

Puzzler - Egg Drop

You are given two identical eggs and access to a 100-story building. You want to determine the highest floor from which an egg will not break when dropped. If an egg is dropped and does not break, it is undamaged and can be dropped again. However, once an egg is broken, that’s it for that egg.

If an egg breaks when dropped from floor n, then it would also have broken from any floor above that. If an egg survives a fall, then it will survive any fall shorter than that.

The question is: What strategy should you adopt to minimize the number egg drops it takes to find the solution? (And what is the worst case for the number of drops it will take?)

Solution: Suppose we start by dropping the first egg from floor 50 and it breaks. The only way to determine the correct floor is to then start with floor 1, then 2, 3, ... until our second egg breaks. If the highest floor is 49 then we get 50 total egg drops -- far from optimal. Generalizing, if our first drop is from floor n and the egg breaks, then in the worst case we require n total egg drops. We know that the solution is between 1 and n-1 inclusive. On the other hand, if the egg does not break, we know the solution is between n+1 and 100. Within this range we can use the same technique but instead drop from n-1 floors above floor n. If the egg then breaks, then in the worst case we still total n egg drops. If it doesn't break we continue by taking our third drop to be n-2 floors above the second drop, and so on.. So assuming the first egg does not break, our successive drops are n, 2n-1, 3n-3, 4n-6, ..., n(n+1)/2. -- given by n, n+(n-1), n+(n-1)+(n-2), ..., n+...+1. When the egg finally does break on the kth drop, we drop the second egg incrementally from the floor of the (k-1)th drop up to the floor of the kth drop and in the worst case we get n total drops. Now we simply need to minimize n, which is done by taking the smallest integer n such that n(n+1)/2 > 99. Our solution is thus 14 total drops.


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