Question

图中的大立方体由 $27$ 个大小相同的小立方体组成。大立方体每个面的相对面都用同样方式涂色。至少有一个面被涂色的小立方体共有多少个？

The large cube shown is made up of $27$ identical sized smaller cubes. For each face of the large cube, the opposite face is shaded the same way. The total number of smaller cubes that must have at least one face shaded is

$10$

$16$

$20$

$22$

$24$

答案：C

知识点：正方体补集计数

难度评级：1200

解答：

先数没有任何涂色面的立方体，再从 $27$ 中减去。一个小立方体未涂色，当且仅当它所有外露的小方格都是空白。

三种面图案如下：顶面和底面只有中心小方格涂色；一对相对侧面是四个角和中心涂色；另一对相对侧面是四条边的中点涂色。

恰有 $7$ 个小立方体避开所有涂色小方格：大立方体最中心隐藏的一个；那对边中点涂色面中心处的两个小立方体，它们唯一外露的小方格是空白中心；以及四个在“角和中心涂色”的面与顶面或底面相交边中点处的小立方体，因为那里每个外露小方格都是空白边格。

因此有 $27 - 7 = 20$ 个小立方体至少有一个涂色面。

所以正确答案是 C。

Count the cubes with no shaded face and subtract from $27.$ A small cube is unshaded exactly when every one of its exposed squares is blank.

The three patterns are: the top and bottom faces show only their center square shaded; one pair of opposite side faces shows the four corners and the center shaded; the remaining pair shows the four edge-midpoints shaded.

Exactly $7$ small cubes avoid every shaded square: the one hidden cube at the very center of the block; the two cubes at the centers of the edge-midpoints faces, whose only exposed square is that blank center; and the four cubes at the midpoints of the edges where a corners-and-center face meets the top or bottom face, since there each exposed square is a blank edge cell.

Hence $27 - 7 = 20$ cubes have at least one shaded face.

Thus, the correct answer is C .

1987 AMC 8 第 7 题

答案：C

难度评级：1200

解答：

其他年份的第 7 题