2020 AMC 10A Problem 10

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 cube geometry sum of first n squares

Difficulty rating: 1420

Small hint:

Start with the surface areas of all seven separate cubes

Big hint:

Each contact hides two faces with the side length of the smaller cube

Video solution:

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Written solution:

The cube side lengths are

1,2,3,4,5,6,7

, stacked from largest on bottom to smallest on top. The sum of the surface areas of the separate cubes is

6(1^2+2^2+\cdots+7^2)

=6\cdot140=840

.

Each contact hides two square faces, with areas

1^2,2^2,\ldots,6^2

. Subtracting these hidden faces gives

840

-2(1^2+2^2+\cdots+6^2)

=840-182=658

. Thus, B is the correct answer.