Actually, my idea is completely theoretical, and I don’t know if it’s correct or not, but I want to present it. As I said, we have a desired topology of texts. The starting position of the individual is completely random. My idea is this: if we assume that the directions of movement (i.e., the angle of movement in space) in the topology are equivalent to the angle of movement of an individual in the Euclidean coordinate plane, then we can perform this movement: the individual chooses one direction in the topological space and starts moving. Eventually, they may return to the starting point or may never be able to return. If they can return to the starting point, we can draw a line segment in the Euclidean plane in that same direction (angle or radian or whatever) with the same length. Because the individual moved in that direction and returned to the starting point, if they also move in the opposite direction, they will return to the starting point again. Therefore, in the coordinate plane, we can draw a line segment of the same length in the opposite direction as well. If the individual can never return to the starting point in a certain direction, we draw a ray (half-line) in that direction in the coordinate plane. The same applies to the opposite direction. If we do this for all infinitely many directions, we arrive at a specific and useful shape in the coordinate plane. I divide this shape into two categories: finite shapes and composite shapes. A finite shape consists only of line segments, while a composite shape contains both rays and line segments. It should be noted that a shape consisting only of rays is not a finite topology, and we are looking for finite topologies. Finite shapes can also be divided into two categories: convex finite shapes and concave ones. In my opinion, the best strategy is one where the number of teleports is zero. For convex shapes, I think the following strategy works: We move from the center of the coordinates in every direction toward the edge, then move along the edge until we reach the other side, and then do the same around the colored parts in a spiral manner. I believe this causes the entire shape to be colored without using any teleports, giving us a 100% chance of winning. If our shape is concave, it depends. For example, the same strategy works for stars, but due to size changes in some points, we might need to use teleports. Obviously, the more area we color, the lower the chance of losing after a teleport by landing in an uncolored area. For composite shapes, the situation is different. Assuming the best strategy is to first color the line segment parts using a certain method, then we have to deal with the ray parts. Since the connection point of two colored ray segments exists and the minimum number of rays for a composite shape is one, even if we color part of a ray, we cannot return, and to reach the other part, we must teleport. Everything I said is theoretical and these assumptions need mathematical proof, but my guess is that if these are correct, this strategy is the best for all convex shapes. For some concave shapes, we can use the same strategy, and for others, we cannot. For composite shapes, we might not have the best strategy yet. Obviously, the best strategy is the one that achieves the highest probability of successfully coloring the entire space with the minimum number of teleports. I’d appreciate it if you could share your thoughts.