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?

A. 35

B. 37

C. 39

D. 43

E. 49

1. 2020 AMC 8 官方解析

The fourth hexagon has 3 bands of dots surrounding a center dot, as shown below. The innermost band has $6 \cdot 1 = 6$ dots, the middle band has $6 \cdot 2 = 12$ dots, and the outermost band has $6 \cdot 3 = 18$ dots. Therefore the total number of dots in the fourth hexagon is $1 + 6 + 12 + 18 = 37$.
第四个六边形中有3个点层围绕着中间一个圆点，如下图所示．最里面的点层有$6 \cdot 1 = 6$个点，中间的点层有$6 \cdot 2 = 12$个点，最外面的点层有$6 \cdot 3 = 18$个点．因此，第四个六边形中总共有$1 + 6 + 12 + 18 = 37$个点．

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 $1 + 2 + 3 = 6$ dots, so the total number of dots in the fourth hexagon is $1 + 6 \cdot 6 = 37$.
除中心点外，每个六边形可细分为六个相等的三角形区域，如下图所示．在第四个六边形中，每个三角形区域包含$1 + 2 + 3 = 6$个点，因此第四个六边形中点的总数为$1 + 6 \cdot 6 = 37$．

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 $7$ dots on a diagonal and $6 + 5 + 4 = 15$ dots on either side, for a total of $7 + 2 \cdot 15 = 37$ dots.
在每个六边形中，对角线上的点形成一条对称线，如下图所示．第四个六边形在对角线上有$7$点，在对角线的任一侧都有$6 + 5 + 4 = 15$个点，总共有$7 + 2 \cdot 15 = 37$个点．