AMC8 2020

AMC8 2020 · Q4

AMC8 2020 · Q4. It mainly tests Basic counting (rules of product/sum), Patterns & sequences (misc).

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 hexagon?

下面展示了三个大小递增的六边形。假设点图案继续，使得每个连续的六边形包含一圈更多的点。下一个六边形有多少个点？

(A) 35 35

(B) 37 37

(D) 43 43

(E) 49 49

Answer

Correct choice: (B)

正确答案：（B）

Solution

Looking at the rows of each hexagon, we see that the first hexagon has $1$ dot, the second has $2+3+2$ dots, and the third has $3+4+5+4+3$ dots. Given the way the hexagons are constructed, it is clear that this pattern continues. Hence, the fourth hexagon has $4+5+6+7+6+5+4=\boxed{\textbf{(B) }37}$ dots.

观察每个六边形的行，我们看到第一个六边形有 1 个点，第二个有 $2+3+2$ 个点，第三个有 $3+4+5+4+3$ 个点。鉴于六边形的构造方式，很明显这种图案会继续。因此，第四个六边形有 $4+5+6+7+6+5+4=\boxed{\textbf{(B) }37}$ 个点。

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