2020 AMC 12A Problem 7

Question

Seven cubes, whose volumes are

1,

8,

27,

64,

125,

216,

and

343

cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the cube below it. What is the total surface area of the tower (including the bottom) in square units?

644

658

664

720

749

Answer: B

Concepts:surface area sum of first n squares

Difficulty rating: 1340

Small hint:

The cubes have side lengths

1, 2, \ldots, 7;

handle the four vertical faces and the horizontal faces separately

Big hint:

Looking straight down, all upward-facing horizontal area equals the base’s

7 \times 7;

the same holds looking up

Solution:

The side lengths are

1, 2, \ldots, 7.

The four side faces of cube

k

contribute

4k^2,

so the vertical faces total

4(1^2 + 2^2 + \cdots + 7^2)

= 4 \cdot 140 = 560.

Viewed from directly above, every upward-facing horizontal patch projects onto the

7 \times 7

base without overlap, giving

49.

Viewed from below, the same is true, giving another

49.

The total surface area is

560 + 49 + 49 = 658.

Thus, B is the correct answer.