AMC 8 1991 Problem 24
AMC8 1991 年第 24 题
立体几何★★★★☆
A cube of edge 3 cm is cut into smaller cubes, not all the same size. If the edge of each of the smaller cubes is a whole number of centimeters, then N=
一个棱长为 3 cm 的正方体被切成 个小正方体，不全部大小相同。如果每个小正方体的棱长都是整数厘米，则 N=
- A.4
- B.8
- C.12
- D.16
- E.20
正确答案 · Correct Answer
E
解析 · Solution
解法1
常规解法 · 体积 + 不可重叠
小立方体棱长是小于 3 的整数，只能是 1 或 2。「不全相同」要求至少一个棱长 2；但两个棱长 2 放不下（2+2>3），故恰好一个。它体积 8，余下 27-8=19 全是单位立方体，共 19 个，N=1+19=20。
巧解 · 体积方程
设棱长 2 的有 个，则总体积 27=8k+(\text{单位立方个数})。几何上最多放 1 个棱长 2，而「不全相同」又要求至少 1 个，故 k=1，单位立方 19 个，N=20。
Solution 1
Elementary · volume + no overlap
Each small cube has integer edge less than 3, so edge 1 or 2. 'Not all the same' forces at least one edge-2 cube, but two edge-2 cubes cannot fit (since 2+2>3), so there is exactly one. It has volume 8, leaving 27-8=19 for the unit cubes, i.e. 19 of them, so N=1+19=20.
Slick · volume equation
By volume, where counts edge-2 cubes. Geometry allows at most one edge-2 cube, and 'not all the same' forces at least one, so k=1, giving 19 unit cubes and N=20.
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