Question

一个图形由单位立方体构成。每个立方体至少与另一个立方体共享一个面。若要得到图中所示的正视图和侧视图，最少需要多少个立方体？

A figure is constructed from unit cubes. Each cube shares at least one face with another cube. What is the minimum number of cubes needed to build a figure with the front and side views shown?

$3$

$4$

$5$

$6$

$7$

答案：B

知识点：立体几何最优化

难度评级：1330

小提示：

尝试只用三个立方体同时满足两个视图

Try to satisfy both views with only three cubes.

大提示：

图中给出了四个立方体的构造；只需排除三个的情况

A four-cube construction is shown; you only need to rule out three.

解答：

正视图需要一个有三个可见位置的 L 形，所以至少需要三个立方体。

假设恰好有三个立方体。正视图中上下相叠的两个立方体必须共享一个面，因此它们处于同一深度。第三个立方体在正视图中位于下方立方体的旁边。若要和前两个立方体中的一个共享面，它也必须和下方立方体处于同一深度；否则它与这两个立方体都不相连。因此三个立方体都在同一个深度列中，侧视图便只有一列，而不是题目要求的 L 形。

图中的四立方体构造具有两个所需视图，所以最小值是 $4$ 。

所以正确答案是 B。

The front view requires an L-shape with three visible positions, so at least three cubes are needed.

Suppose there were exactly three cubes. The two cubes that appear one above the other must share a face, so they have the same depth. The third cube appears beside the lower one. To share a face with either of the first two cubes, it must share the lower cube’s depth as well; otherwise it would be disconnected from both. Thus all three cubes would occupy one depth column, giving a one-column side view instead of the required L-shape.

The shown four-cube construction has both required views, so the minimum is $4.$

Thus, B is the correct answer.

2003 AMC 8 第 15 题

答案：B

难度评级：1330

小提示：

大提示：

解答：

其他年份的第 15 题