Two glass balls

Summation
You have two glass balls and you are standing in a 100-story building. If you throw a glass ball out of the window, it will not break if the floor number is less than X. However, it will always break if the floor number is equal or greater than X. You want to determine which floor has number X. What is the minimum number of drops for the worst case scenario to achieve your goal?

Hint

Design a strategy in which it is possible to use the summation rule.

Answer

Intuitively, you want the balls to cover at least all 100 floors. First, we have to come up with a strategy. Suppose you find the number throws. A strategy could be to start at the -th floor. Then, we have the following two scenarios:
  1. The first ball breaks. Since you don't want to take the risk of breaking the second ball (because in that case you won't find the floor number ), you go to the first floor and you go up one level every time until the ball breaks.
  2. The first ball does not break. You go up to floor number . Here, we have the following two scenarios:
    1. In case the ball breaks here, you go to floor and you go up one level every time until the ball breaks.
    2. In case the ball does not break, you go up to level .
Continuing this strategy, the number of total throws will never result in more than throws. To find , we need to solve the following equation: or . Solving this equation results in .