How to solve this problem using Pigeonhole Principle?
A worker plans to work 60 hours in the next 37 days, and he works at least 1 h/day, show that he will be working 13 hours in total in some continuous days.
So there are 23 hours more to spend on 37 days, but how is that related to 13 on continuous days?
Denote the total amount of the worker has worked from the first day until the $n-$th day by $S_n$.
Now $S_n\leq 60$ and $A_n=S_n+13 \leq 73$ for all $n$.
Both $S_n,A_n$ are between $1$ and $73$ and there are $74$ of them. Consequently, by the pigeonhole principle, there are two of them that are equal.
From your condition that the worker works at least one hour per day the collection of $S_n$ are pairwise distinct and the same holds for the $A_n$ family. Therefore there are $n_1$ and $n_2$ such that $A_{n_1}=S_{n_2}$ whence $S_{n_2}-S_{n_1}=13$.
So we conclude that the desired time interval where the worker worked for exactly $13$ hours is between days $n_1+1, n_1+2, \cdots, n_2$