Suppose we have a beach of length $1-$km. Suppose one Day $0$, the beach is empty.
One day $1$, a family comes and puts their umbrella at some point in the beach. This point is fixed forever and cannot be moved.
On day $2$, a new family comes and puts their umbrella in a new coordinate. This umbrella is now fixed forever. However, we are guaranteed that the intervals $[0,\frac{1}{2}], [\frac{1}{2},1]$ each have $1$ umbrella; the old and the new one.
In general, on day $i$, a new family comes in and fixes a new umbrella. We're guranteed that the interval $[0,\frac{1}{i}], ..., [1-\frac{1}{i}, 1]$ all have one umbrella each.
The question is, what is the maximum day $t$ such that we cannot add a new umbrella where the $t$ sub intervals would have $1$ umbrella each?
The question answer say $18$, but I don't see how this is true?! Shouldn't we be able to keep going forever?