Track / Stage 4 / 284 of 340 #544 of 1964

Problem 544

AMC 12 late, AIME early

Combinatorics Difficulty 5.0 Multiple choice

$35 \cdot 49$ Divide a cube with an edge length of $3 \mathrm{~cm}$ into $n$ smaller cubes of different volumes. If the edge lengths of the smaller cubes are whole centimeters, then $n$ is

Pick one

Official solution

[Solution] The volume of a cube with a side length of $3 \mathrm{~cm}$ is $27 \mathrm{~cm}^{3}$ . It can only be divided into one cube with an edge length of $2 \mathrm{~cm}$ , whose volume is $8 \mathrm{~cm}^{3}$ .

Thus, the remaining $19 \mathrm{~cm}^{3}$ can only be divided into cubes with an edge length of $1 \mathrm{~cm}$ , totaling 19 pieces. Therefore, it can be divided into a total of 20 small cubes that are not all the same.
Hence, the answer is $(E)$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.