This is a problem about polygons and their midpoints, so I will try to use some geometry and algebra to solve it. Now, I want to relate the centroid of $P_1$ to the centroid of $P_2$. Therefore, the centroid of $P_2$ is the same as the centroid of $P_1$, which means it also has an $x$-coordinate of $3$. Similarly, the centroid of $P_3$ is the same as the centroid of $P_2$, so it also has an $x$-coordinate of $3$. Now, I want to find the sum of the $x$-coordinates of the vertices of $P_3$. I know that the sum of the $x$-coordinates of the vertices of a polygon is equal to the number of vertices times the $x$-coordinate of the centroid. So, the sum of the $x$-coordinates of the vertices of $P_3$ is $33 \times 3 = 99$.


A $33$-gon $P_1$ is drawn in the Cartesian plane.  The sum of the $x$-coordinates of the $33$ vertices equals $99$.  The midpoints of the sides of $P_1$ form a second $33$-gon, $P_2$.  Finally, the midpoints of the sides of $P_2$ form a third $33$-gon, $P_3$.  Find the sum of the $x$-coordinates of the vertices of $P_3$.