AMC 8 2020 Problem 4
AMC8 2020 年第 4 题
数列与规律★★☆☆☆
Three hexagons of increasing size are shown below. Suppose the dot pattern continues so that each successive hexagon contains one more band of dots. How many dots are in the next hexagonal?
三个面积逐渐增大的六边形如下图所示。假设这种点阵模式以此规律继续，使得后一个六边形比前一个多一层点带。那么图中下一个六边形中有多少个点？
- A.35
- B.37
- C.39
- D.43
- E.49
正确答案 · Correct Answer
B
解析 · Solution
Solution 1
The fourth hexagon has 3 3 bands of dots surrounding a center dot, as shown below. The innermost band has 6 ⋅ 1 = 6 6⋅1=6 dots, the middle band has 6 ⋅ 2 = 12 6⋅2=12 dots, and the outermost band has 6 ⋅ 3 = 18 6⋅3=18 dots. Therefore the total number of dots in the fourth hexagon is 1 + 6 + 12 + 18 = 37 1+6+12+18=37 𝑂 𝑅 OR Excluding the center dot, each hexagon can be subdivided into six congruent triangular regions, as shown in the figure below. In the fourth hexagon, each triangular region contains 1 + 2 + 3 = 6 1+2+3=6 dots, so the total number of dots in the fourth hexagon is 1 + 6 ⋅ 6 = 37 1+6⋅6=37. 𝑂 𝑅 OR In each hexagon, the dots that lie on a diagonal form a line of symmetry, as shown in the figure below. The fourth hexagon has 7 7 dots on a diagonal and 6 + 5 + 4 = 15 6+5+4=15 dots on either side, for a total of 7 + 2 ⋅ 15 = 37 7+2⋅15=37 dots. 𝑁 𝑜 𝑡 𝑒 Note: The number of dots in each hexagon is called a ℎ 𝑒 𝑥 hex number. Generalizing from the first and second solutions, the 𝑛 𝑡 ℎ nth ℎ 𝑒 𝑥 hex number can be calculated as 1 + 6 ⋅ ( 1 + 2 + 3 +... + ( 𝑛 − 1 ) ) = 1 + 6 ⋅ ( 𝑛 ( 𝑛 − 1 ) ) 2 = 1 + 3 𝑛 ( 𝑛 − 1 ). 1+6⋅(1+2+3+...+(n−1))=1+6⋅ 2 (n(n−1)) =1+3n(n−1). Generalizing from the third solution, the nth hex number equals ( 2 𝑛 − 1 ) + 2 ⋅ ( ( 2 𝑛 − 2 ) + ( 2 𝑛 − 3 ) +... + 𝑛 ) = 1 + 3 𝑛 ( 𝑛 − 1 ). (2n−1)+2⋅((2n−2)+(2n−3)+...+n)=1+3n(n−1). It can be shown that the sum of the first in hex numbers is 𝑛 3 n 3. For example, the sum of the first 2 2 hex numbers is 1 + 7 = 8 = 2 3 1+7=8=2 3 the sum of the first 3 3 hex numbers is 1 + 7 + 19 = 27 = 3 3 1+7+19=27=3 3, and the sum of the first 4 4 hex numbers is 1 + 7 + 19 + 37 = 64 = 4 3 1+7+19+37=64=4 3. Answer: 𝐵 B .
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