Question

一个边长为 $3$ 厘米的正方体被切成 $N$ 个较小的正方体，且这些小正方体不全同样大小。如果每个小正方体的边长都是整厘米数，那么 $N =$

A cube of edge $3$ cm is cut into $N$ smaller cubes, not all the same size. If the edge of each of the smaller cubes is a whole number of centimeters, then $N =$

$4$

$8$

$12$

$16$

$20$

答案：E

知识点：正方体体积

难度评级：1140

解答：

这个 $3 \times 3 \times 3$ 正方体的体积是 $27$ 。由于小正方体更小且边长为整数，它们的边长只能是 $1$ 或 $2$ 。原正方体内不能放下两个互不相交的边长为 $2$ 的正方体，因为要将它们分开，某个方向上的总跨度至少为 $4$ 。因此最多只能放下一个边长为 $2$ 的正方体。

剩余体积 $27 - 8 = 19$ ，由 $19$ 个单位正方体填满。总数是 $1 + 19 = 20$ 个正方体。

所以正确答案是 E。

The $3 \times 3 \times 3$ cube has volume $27.$ Since the cubes are smaller and have whole-number edge lengths, their edges are $1$ or $2$ . Two edge- $2$ cubes cannot be disjoint inside the original cube: separating them would require a total span of at least $4$ in some direction. Thus at most one edge- $2$ cube fits.

The remaining $27 - 8 = 19$ of volume is filled by $19$ unit cubes. That is $1 + 19 = 20$ cubes.

Thus, the correct answer is E .

1991 AMC 8 第 24 题

答案：E

难度评级：1140

解答：

其他年份的第 24 题