1987 AMC 8 第 7 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。
7.
图中的大立方体由 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 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
小提示：
小立方体只有在某个外露小方格被涂色时才有涂色面；先数完全未涂色的小立方体，包括最中心隐藏的那个
A small cube shows a shaded face only when one of its surface squares is shaded, so count the cubes that stay completely blank, beginning with the single hidden cube at the center of the block
大提示：
除中心小立方体外，找出所有外露方格都落在空白位置的小立方体，再从 中减去未涂色数
Besides the hidden center cube, find the cubes whose every exposed square lands on a blank position, then subtract the number of blank cubes from
解答：
先数没有任何涂色面的立方体，再从 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 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 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 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 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 cubes have at least one shaded face.
Thus, the correct answer is C .
其他年份的第 7 题
- 1985 AMC 8
- 1986 AMC 8
- 1988 AMC 8
- 1989 AMC 8
- 1990 AMC 8
- 1991 AMC 8
- 1992 AMC 8
- 1993 AMC 8
- 1994 AMC 8
- 1995 AMC 8
- 1996 AMC 8
- 1997 AMC 8
- 1998 AMC 8
- 1999 AMC 8
- 2000 AMC 8
- 2001 AMC 8
- 2002 AMC 8
- 2003 AMC 8
- 2004 AMC 8
- 2005 AMC 8
- 2006 AMC 8
- 2007 AMC 8
- 2008 AMC 8
- 2009 AMC 8
- 2010 AMC 8
- 2011 AMC 8
- 2012 AMC 8
- 2013 AMC 8
- 2014 AMC 8
- 2015 AMC 8
- 2016 AMC 8
- 2017 AMC 8
- 2018 AMC 8
- 2019 AMC 8
- 2020 AMC 8
- 2022 AMC 8
- 2023 AMC 8
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