2016 AMC 12A Problem 10

Question

Five friends sat in a movie theater in a row containing

5

seats, numbered

1

to

5

from left to right. (The directions “left” and “right” are from the point of view of the people as they sit in the seats.) During the movie Ada went to the lobby to get some popcorn. When she returned, she found that Bea had moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up?

1

2

3

4

5

Answer: B

Concepts:invariant logical deduction

Difficulty rating: 1410

Small hint:

The total rightward movement must equal the total leftward movement

Big hint:

Bea moves

+2,

Ceci moves

-1,

and Dee and Edie cancel; find Ada’s shift

Solution:

The net displacement of all five friends is zero. Dee and Edie swapped seats, so their movements cancel. Bea moved

+2

and Ceci moved

-1,

a net of

+1,

so Ada must move

-1

to balance.

Ada returns to an end seat; since she moved one seat to the left, that seat must be seat

1,

so she had been sitting in seat

2.

Thus, the correct answer is B.