Question

从边长为 $3$ 的实心立方体的每个角上各去掉一个边长为 $1$ 的实心小立方体。剩下的立体有多少条棱？

A solid cube of side length $1$ is removed from each corner of a solid cube of side length $3.$ How many edges does the remaining solid have?

$36$

$60$

$72$

$84$

$108$

去掉角上的小立方体不会消除原来大立方体的十二条外边界棱，只会增加新的棱。

每去掉一个角，会新增 $9$ 条棱。

共有 $8$ 个角，所以新增 $8 \cdot 9 = 72$ 条棱。再加上原来的 $12$ 条棱，总数为 $72 + 12 = 84$ 。

所以正确答案是 D。

Removing the cubes does not remove any edges from the original cube. It only adds edges.

After removing each cube, we can see that $9$ extra edges are added to the solid.

$8$ cubes are removed, which means $8 \cdot 9 = 72$ edges are added to the original $12$ edges, for a total of $72 + 12 = 84$ edges.

Thus, D is the correct answer.