PROBLEM $2$ SAFETY

For an engineering class project, you are assigned to create a container that can safely allow water

balloons to fall from as high as possible. Part of the assignment is to test the container. Suppose

your campus has a special building, built for the purpose $($this is a very advance Engineering

campus, with buildings built to contain special sensors for water balloon testing$).$ This building

has $36$ floors. Now, ideally your device would let the water balloon fall from $36$ floors without

breaking, but very few students in history have ever made a container that works at this height$).$

There are some rules to the testing, as follows.

If the water balloon survives a drop, then it can be used again for another test.

But, if the water balloon breaks, then of course it cannot be used again.

If a balloon would break when dropped from a given floor, then it would also break if

dropped from a higher floor.

If a balloon can survive when dropped from some floor, then it would also be safe to drop

from a lower floor.

It is possible that your container will break the balloon even when dropped from the first floor,

or it is possible that it will work even for floor $36.$ Now, if you only had one balloon, then you

really could not do anything other than start from Floor $1$ and work your way up sequentially.

But, assume that you have two water balloons available for testing. You must devise some testing

order that allows you to determine the lowest floor at which the balloon will break, using the

least number of balloon drops in the worst case, no matter which floor happens to be the lowest

breaking floor.