Jessica the Combinatorics Student, part 2 The original question about Jessica, which I encourage review of, is as follows:

Jessica is studying combinatorics during a $7$-week period. She will study a positive integer number of hours every day during the $7$ weeks (so, for example, she won't study for $0$ or $1.5$ hours), but she won't study more than $11$ hours in any $7$-day period. Prove that there must exist some period of consecutive days during which Jessica studies exactly $20$ hours.

My follow-up question is this: 

How far into the course (how many days) do you need before you can be sure that Jessica has had some sequence of consecutive days with a twenty-hour study total?

My suspicion is that the answer is: twenty days. I think my answer at the original question can be used to demonstrate that four weeks is certainly enough
 A: Here's code to run an exhaustive search, which confirms your suspicion that the answer is $20$ days.
Interestingly, it's still $20$ days if Jessica is allowed to work up to $14$ hours in any $7$-day period. If she's allowed to work $15$ hours, a repeating pattern of $5,1,1,1,5,1,1,1,\ldots$ avoids any sums of $20$ for any number of days.
A: I think this might essentially be a reformulation of Wiley's proof in the case when at most $13$ hours of study are permitted over a $7$-day period.  However, the application of the pigeonhole principle is perhaps simpler.
For $0 \le i \le 20$, let $S_i$ be the total number of hours studied by the end of the $i$th day (setting $S_0 = 0$), and consider these values modulo $20$.
Since we have $21$ terms taking $20$ possible values, there are some $0 \le i < j \le 20$ such that $S_i = S_j$.  It follows that the total number of hours of study between days $i+1$ and $j$ (inclusive) is a multiple of $20$.
If it is not exactly equal to $20$ hours, then it must be at least $40$ hours.  However, this is over a span of at most $20$ days.  Dividing this into three periods of at most $7$ days (say the first week, second week, and third week), by averaging we find that she must have worked at least $14$ hours during one of the weeks, which is not allowed.
Thus she must actually have studied exactly $20$ hours between days $i+1$ and $j$.
