Math Teacher.
The fastest way to find out which floor the egg breaks (or minimize the worst case scenario of number of trials) is to drop the first egg at the floor that equals the square root of how many floors are left. For example, in the beginning, when there are 100 floors, the most efficient floor to drop the first egg from is floor number 10, because the square root of 100 = 10.
Assuming the egg doesn't break on floor 10, you would drop the "first" egg again from the floor equal to the square root of 90, because there are now 90 floors left. And so on and so on. Then the "first" egg breaks, you would drop the second egg one floor at a time starting from the previous threshold. If you do it this way, you would be able to decrease the worst case scenario by 1 from if you dropped the "first" egg by 10s every time. The more floors you begin with, the greater this method has in increasing efficiency.
The reason the square root is always the best choice can be thought of using a rectangle. We are trying to minimize the width and length (which correspond to egg one and egg two trials). Since we have 100 floors, which is represented by the area of the rectangle, the width and length can both be 10 (square root), or 50 and 2, or 25 and 4, etc.. The worst case scenario for each of these examples would be the sum of the two sides minus 1; for 10 and 10, the worst case scenario would be 19 trials to make it to the 99th floor. for 50 and 2, the worst case scenario would be 51 trials to make it to the 99th floor. As you can see, the square root width and length always gives the best worst case scenario.