1998 AMC 12 第 27 题
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所有题目均经美国数学协会（MAA）官方合法授权使用。
27.
一个 的大立方体由二十七个 的小立方体组成。按如下方式给大立方体“打隧道”：首先，如图所示，移除构成每个面中心的六个 小立方体以及正中心的一个 小立方体。其次，对剩余二十个 小立方体逐个进行同样的操作，即移除各面中心的单位立方体以及正中心的单位立方体。最终图形的表面积为
A cube is composed of twenty-seven cubes. The big cube is “tunneled” as follows: First, the six cubes which make up the center of each face as well as the center cube are removed as shown. Second, each of the twenty remaining cubes is diminished in the same way. That is, the center facial unit cubes as well as each center cube are removed. The surface area of the final figure is
答案：E
小提示：
第一步之后，把剩余二十个大块小立方体分成角块和棱块两类
After the first stage, classify the twenty remaining large subcubes as corner or edge cubes
大提示：
第二步每打通一个隧道，都要减去外露的中心正方形，并加上新露出的隧道内壁
For each second-stage tunnel, subtract exposed center squares and add the newly exposed tunnel walls
解答：
第一步之后， 个角块各贡献 个外露单位面， 个棱块各贡献 个。给一个角块打隧道会移除 个外露单位正方形，并增加 个隧道内壁正方形；给一个棱块打隧道会移除 个外露面，也增加 个内壁面。因此最终面积为 所以正确答案是 E。
After the first stage, corner subcubes contribute exposed units each, and edge subcubes contribute each. Tunneling a corner subcube removes exposed unit squares and adds tunnel-wall squares; tunneling an edge subcube removes and also adds Hence the final area is Thus E is correct.
其他年份的第 27 题
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- 1951 AMC 12
- 1952 AMC 12
- 1953 AMC 12
- 1954 AMC 12
- 1955 AMC 12
- 1956 AMC 12
- 1957 AMC 12
- 1958 AMC 12
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- 1968 AMC 12
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- 1970 AMC 12
- 1971 AMC 12
- 1972 AMC 12
- 1973 AMC 12
- 1974 AMC 12
- 1975 AMC 12
- 1976 AMC 12
- 1977 AMC 12
- 1978 AMC 12
- 1979 AMC 12
- 1980 AMC 12
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- 1987 AMC 12
- 1988 AMC 12
- 1989 AMC 12
- 1990 AMC 12
- 1991 AMC 12
- 1992 AMC 12
- 1993 AMC 12
- 1994 AMC 12
- 1995 AMC 12
- 1996 AMC 12
- 1997 AMC 12
- 1999 AMC 12