With 2 eggs. Drop egg #1 from floor 18. If egg #1 breaks use egg #2 starting from the ground up to possibly floor 17, being possibly 18 drops. If egg #1 survives you could just work upwards, possibly another 18 drops, or 19 in total, but you don't need to as you can again split and go to floor 27, but the worst case is the broken egg #1 from floor 18, so the answer is 18.

With 3 eggs. Drop egg #1 from floor 24. If egg #1 breaks use egg #2 from floor 11. If egg #2 breaks, use egg #3 from ground up being possibly 10 drops. If egg #2 doesn't break you still have 2 eggs and the answer is between floors 12 and 23. If egg #1 doesn't break you still have 3 eggs and the answer is between floors 25 and 36. The worst case is egg #2 broken from floor 11 so the answer is 12. (or maybe 10)

With 4 eggs. Drop egg #1 from floor 27. If it breaks drop egg #2 from floor 13. If it breaks drop egg #3 from floor 7. If it breaks then there are possibly 6 drops with egg #4 from the ground up. The answer is 9. (or maybe 6 or 7)

The pattern appears to suggest the answer is the number of floors divided by the number of eggs (or not). But is it?

I regret that neither of your hypotheses are correct, Livo, although "start from floor 18" is where most analyses commence (at least, for those who have not encountered the problem previously). What you will find, when you investigate "start from floor 18" further, is that there is another "natural" starting point, and when you investigate that one you will find that it too is sub-optimal. Which will cause you to think "OK, if there is no 'natural' starting floor that is optimal, clearly the starting floor (and subsequent floors) must be found in another way". And then you will be

*en route * to the solution. As to three eggs or more, that one took me a further day to find, but it is now found and the method will generalise to $n$ eggs, for $n <= floors$.

Only one person (in my circle of correspondents) has come up with the correct answer for two eggs, and he admitted he had seen the problem before and had solved it by writing a computer program ! He also offered an answer for three eggs, which was wrong, but the error

*could * have been no more than a simple typo.

** Phil.