2. We consider dotted hexagons with $1,2,3, \ldots$ dots on one side, see also the figure below. The number of dots in such a hexagon is called a hexagonal number. The first hexagonal number is 1, the second is 7, and the third is 19. Which of the following numbers is also a hexagonal number?
A) 81
B) 128
C) 144
D) 169
E) 187

![](https://cdn.mathpix.com/cropped/2024_04_17_01cb006263f00c7722c1g-1.jpg?height=326&width=171&top_left_y=1533&top_left_x=1499)
A2. D) 169 Suppose we start with the first hexagon (a single dot). Then we can get the next hexagon by placing six new dots around the dot. Subsequently, we can get the third hexagon by adding another layer around the current hexagon. In this case, we add twelve new dots. We observe that the number of dots added to the edge to get the next hexagon increases by six each time. This is because the number of dots on each side of the hexagon increases by one.

![](https://cdn.mathpix.com/cropped/2024_04_17_fc21619166b069277c26g-1.jpg?height=317&width=163&top_left_y=909&top_left_x=1666)

After 19, the next hexagonal number is thus $19+18=37$, then we get $37+24=61$, then $61+30=91$, then $91+36=127$, then $127+42=169$, and finally $169+48=217$. We see that of the five given numbers, only 169 is a hexagonal number.