Three Little Pigs

Recall that after defeating the wolf, all three little pigs settled at the third pig's house. During the long winter evenings in their cozy stone home, they would read books and play games. However, as winter dragged on, even their favorite games became tiresome. So, the pigs decided to invent a new board game, similar to checkers or chess, but designed for three players. They drew a triangular board and placed an equal number of points at equal intervals along the sides of the triangle. Then, they connected these points on adjacent sides with segments parallel to the third side. This resulted in a triangular field made up of smaller triangular cells (a small section of which is shown in the picture).

The pigs realized that they often argued during games, so they decided that moves in their new game should be recorded. To do this, they needed to number the triangles and their vertices. The pigs numbered the vertices, choosing one as the starting point (as shown in rectangular frames in the picture), and numbered the triangles (as shown in circles in the picture).

The third pig noticed that the vertex numbers of the triangular field could be easily determined by the triangle's number. Therefore, when recording moves, there was no need to write down the vertex numbers. For example, triangle $7$ has vertices $5$, $8$, and $9$, triangle $11$ has vertices $7$, $8$, and $12$, and so on. However, the first and second pigs did not immediately grasp this idea, so they needed a program that calculates the vertex numbers of a triangle based on its number.