AMC 8 2020 Problem 9
AMC8 2020 年第 9 题
立体几何★★☆☆☆
Akash's birthday cake is in the form of a inch cube. The cake has icing on the top and the four side faces, and no icing on the bottom. Suppose the cake is cut into 64 smaller cubes, each measuring inch, as shown below. How many small pieces will have icing on exactly two sides?
Akash 的生日蛋糕是一个 4ｘ4ｘ4 的正方体（边长单位：英寸）。这个立方体的顶部和 4 个侧面都有冰覆盖，但底而没有。假设这个正方体被分成 64 个 1ｘ1ｘ1 英寸的小正方体，如下图所示。那么这些小正方体中有多少个恰好两面有冰？
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正确答案 · Correct Answer
D
解析 · Solution
Solution 1
Notice that, for a small cube which does not form part of the bottom face, it will have exactly 2 faces with icing on them only if it is one of the 2 center cubes of an edge of the larger cube. There are 12-4 = 8 such edges (as we exclude the 4 edges of the bottom face), so this case yields small cubes. As for the bottom face, we can see that only the 4 corner cubes have exactly 2 faces with icing, so the total is 16+4 = 20.
The following diagram shows 12 of the small cubes having exactly 2 faces with icing on them; that is all of them except for those on the hidden face directly opposite the front face. But the hidden face is an exact copy of the front face, so the answer is 12+8=20.
It is observable that only the middle-edged pieces of each face will have the icing exactly on two sides. There are 4 such pieces on each face. Considering 5 faces of the cube (since the bottom is not iced), we can state that the number of such pieces of dimensions 1x1x1 will be 5x4=20. Therefore, our answer is 20. by WrenMath On a 4 x 4 Rubik's cube, there are exactly 24 'edge' pieces, 8 'corners', and 24 'center' pieces. Edge pieces have 2 frosted faces (the ones on the bottom only have one, corners have 3 frosted faces, and centers have 1. So since we have 24 edges pieces, we subtract the 8 'edge' pieces on the bottom (they only have one frosted face), and then we add the 4 bottom 'corner' pieces (they also have frosted faces). we get 24-8+4=20
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