You have a 100 story building and two eggs. These are especially strong eggs. There is some floor below which the egg will not break if dropped.

You have a 100 story building and two eggs. These are especially strong eggs. There is some floor below which the egg will not break if dropped. What is the worst case upper bound on the number of drops you must make to determine this floor?

Let N the number of drops you need to find the first floor that breaks eggs. Go to the Nth floor and drop an egg. If it breaks you have N - 1 more drops to test the N - 1 floors below. If it doesn't breaks, go to the N + (N - 1)th floor and drop the same egg. If it breaks you have N -2 drops left to test the N - 2 floors between Nth floor and 2N -1 floor. By a similar analysis you approach the top of the building with
N + (N - 1) + (N - 2) + (N - 3) + . . . + 1 <= 100
N ( N + 1 ) / 2 <= 100
This is a quadratic equation which yields N <= 13.65. If N = 13 you can only analyze a building of 91 floors. It takes N = 14 to test a 100 floor building. Final solution. You go to the 14th floor and drop an egg. If it breaks you have 13 more drops to test floors 1 to 13. If it doesn't break you go to the 27th (14 + 13) and drop the first egg again. If it breaks you have 12 drops left to test the 12 floors above 14 and below 27. You continue up the building until you reach the 99th floor (14+13+12+11+10+9+8+7+6+5+4) with three drops left. If it breaks you have three drops left to test the three floors between the 95th floor and the 99th floor.

3 eggs: for first egg, we use binary search, for the rest two eggs we compute like N(N+1)<=50, so maximum time is 10+1(not optimized )

N floors, m eggs, l drops
C(l,1)+C(l,2)+…+C(l,m)<n