2016 AMC 10B Problem 4

Give each day a number. Call the Monday when she finished book $1$ "day $1$."

1. Book $2$ takes $2$ more days, so it is finished on day $1 + 2 = 3$. Day $3$ is two days after Monday, a Wednesday. That matches the problem, so the numbering is right.
2. Keep going. Book $3$ ends on day $1 + 2 + 3 = 6$, book $4$ on day $1 + 2 + 3 + 4 = 10$, and so on. Book $15$ ends on day
$$1 + 2 + 3 + \cdots + 15 = \frac{15 \cdot 16}{2} = 120.$$
3. Day $120$ is $120 - 1 = 119$ days after day $1$.
4. The weekday repeats every $7$ days, and $119 = 7 \cdot 17$. So $119$ days is exactly $17$ weeks, and we land on the same weekday as day $1$.

The answer is $\boxed{\textbf{(B)}\ \text{Monday}}$.

"Finished book $n$" means "the total of all reading days so far," so the finishing days are the running totals $1, 3, 6, 10, \ldots$. For the weekday, only the remainder when you divide the number of days between two dates by $7$ matters. Be careful to count the days between the dates, which is one less than the day number.

Number the days so that day $1$ is the Monday on which book $1$ is finished. Book $2$ takes two more days and is finished on day $3$, a Wednesday, which matches the statement.

Book $n$ is finished on day $1 + 2 + \cdots + n$, so book $15$ is finished on day
$$1 + 2 + \cdots + 15 = \frac{15 \cdot 16}{2} = 120.$$

Day $120$ is $119$ days after day $1$. Since $119 = 7 \cdot 17$, that is exactly $17$ weeks later, so day $120$ falls on the same weekday as day $1$.

The answer is $\boxed{\textbf{(B)}\ \text{Monday}}$.