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2016 AMC 10B Problem 4

2016 AMC 10B Problem 4AMC 10 Step by Step

Name: Date: September 13, 2026

Clocks, Calendars & Time · difficulty 2 of 5 · about 2 min · position P1-10

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?

A)
$\text{Sunday}$
B)
$\text{Monday}$
C)
$\text{Wednesday}$
D)
$\text{Friday}$
E)
$\text{Saturday}$

Answer

Key insight

Book 15 is finished on day 1+2+...+15 = 120; that is 119 = 7*17 days after the Monday of day 1, so also a Monday.

Solution

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

Why this works

"Finished the $n$ -th book" translates to a cumulative total, the triangular number $n(n+1)/2$ . Weekday questions are then arithmetic mod $7$ ; the only care needed is the fencepost: the gap between day $1$ and day $120$ is $119$ days, not $120$ . The Wednesday clue is just a consistency check.

The trap

Counting 120 days forward from the Monday (landing on Tuesday) instead of 119, because day 1 itself is the Monday.

Common mistakes

Counting $120$ days forward from the Monday (landing on Tuesday) instead of $119$ , because day $1$ itself is the Monday.
Summing only $2 + 3 + \cdots + 15 = 119$ and then also adding $119$ to a Monday, or forgetting the first book's day entirely.

Techniques

Set up the equation/formula and compute; no special trick needed