[GGG]

For the ball drop

Spoiler!

I assume this has been solved by others but before I look the balance appears to be having the hundredth floor break your egg needing 99 floor to be successful and the n-1 floor after any drop and having to test from your last successful drop.

So you need to start at whatever floor exceeds the floor number by the least decreasing the drop spacing by 1 each time. So checking with the (N+1)*N/2 is least over 100 so 13 is 91 and 14 is 105. So I think max it will take is 14 drops.