18. Prince Gvidon had three sons. Among his descendants, 93 each had two sons and no daughters, while all the others died childless. How many descendants did Prince Gvidon have in total?
18. Each of the descendants who have children adds two more descendants to the three descendants of the first generation. Since 93 descendants had children, this added $2 \cdot 93 = 186$ descendants, making a total of $3 + 186 = 189$. This problem ad-

![](https://cdn.mathpix.com/cropped/2024_05_21_7d0959e1f193b6338b84g-34.jpg?height=457&width=571&top_left_y=343&top_left_x=0)
a)

mits a beautiful geometric interpretation. Let's draw the family tree of Prince Gvidon (Fig. 18). From each vertex of this tree, either two segments or none emerge. The vertices of the first type are circled in black. The total number of descendants is equal to the number of segments in the tree. But each segment (except the three upper ones) emerges from a circled

![](https://cdn.mathpix.com/cropped/2024_05_21_7d0959e1f193b6338b84g-34.jpg?height=438&width=420&top_left_y=935&top_left_x=589)
b)

Fig. 17.

black vertex (of which there are 93). Therefore, the total number of segments is $2 \cdot 93 + 3 = 189$

3 E. B. Dynkin et al.