2016 AMC 10B Problem 4

P1-10~2 minPrint

Clocks, Calendars & Time Modular Arithmetic Sequences & Series Set up the equation/formula and compute; no special trick needed

Zoey read $15$ books, one at a time. The first book took her $1$ day to read, the second book took her $2$ days to read, the third book took her $3$ days to read, and so on, with each book taking her $1$ more day to read than the previous book. Zoey finished the first book on a Monday, and the second on a Wednesday. On what day of the week did she finish her $15$ th book?

Show solution without answering

Key insight

Book 15 is finished on day 1 + 2 + ... + 15 = 120. Day 120 is 119 days after day 1, and 119 is exactly 17 weeks.

Solution

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

Why this works

"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.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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

Review in the AoPS books

Introduction to Algebra, Chapter 21: Sequences & Series (§21.1, §21.2, §21.3, §21.4)
Introduction to Number Theory, Chapter 12: Introduction to Modular Arithmetic
Intermediate Algebra, Chapter 17: More Sequences and Series (§17.1, §17.2)
Introduction to Number Theory, Chapter 10: Units Digits