AMC 8 2001 Problem 24
AMC8 2001 年第 24 题
基本计数原理★★★★☆
Each half of this figure is composed of 3 red triangles, 5 blue triangles and 8 white triangles. When the upper half is folded down over the centerline, 2 pairs of red triangles coincide, as do 3 pairs of blue triangles. There are 2 red-white pairs. How many white pairs coincide?
该图形的每一半由 3 个红色三角形、5 个蓝色三角形和 8 个白色三角形组成。当上半部分沿中心线向下折叠时，2 对红色三角形重合，3 对蓝色三角形重合。还有两对红－白对。那么有多少对白色三角形重合？
- A.4
- B.5
- C.6
- D.7
- E.9
正确答案 · Correct Answer
B
解析 · Solution
解法1
常规解法 · 逐类扣减
每半有红 3、蓝 5、白 8。2 对红重合用掉每侧 2 红，剩红 1；3 对蓝用掉每侧 3 蓝，剩蓝 2；2 对红白用掉每侧 1 红 1 白（红恰用完），剩蓝 2、白 7；剩下的蓝只能配白（再多蓝对就超 3 对），成 2 对蓝白、用掉每侧 2 白，剩白 5，两两相配得 5 对白。
巧解 · 数白色去向
每侧 8 白，按配对去向分账：红白对每侧用 1 白；每侧剩余 2 个蓝只能配白，用 2 白；其余白两两相配设 对、每侧用 白。于是 1+2+w=8，w=5。
Solution 1
Elementary · subtract by cases
Each half has 3 red, 5 blue, 8 white. The 2 red-red pairs use 2 red per side (1 red left); the 3 blue-blue pairs use 3 blue per side (2 blue left); the 2 red-white pairs use the last red and 1 white per side (0 red, 7 white left). The 2 remaining blue per side must pair with white, using 2 white per side and leaving 5 white per side, which pair up as 5 white-white pairs.
Slick · count where the white goes
Count whites per side (8 each) by where they go: the red-white pairs use 1 white/side; the 2 leftover blue per side must pair with white, using 2 white/side; the rest form white-white pairs, per side. Then 1+2+w=8, so w=5.
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