AMC 8 1987 Problem 7
AMC8 1987 年第 7 题
立体几何★★☆☆☆
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
所示大正方体由 27 个相同大小的小正方体组成。大正方体的每个面，其相对面的着色方式相同。至少有一个面被着色的小正方体总数是
- A.10
- B.16
- C.20
- D.22
- E.24
正确答案 · Correct Answer
C
解析 · Solution
Solution 1
Notice that no two shaded tiles are adjacent. Therefore, each small cube has at most one face shaded, and the number of small cubes with at least one face shaded is the same as the number of shaded tiles on the entire cube. Since there are 10 visible shaded tiles, and the hidden faces of the entire cube are identical to the shown faces, there must be small cubes with at least one face shaded.
The top and bottom layers each have 2 cubes with no shaded sides, which are located in the middle of the front and back rows. The middle layer has 3 cubes with no shaded sides, which are all in the middle row. Therefore, cubes have no shaded sides, and 27 - 7 = 20 cubes have at least one shaded side.
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