A spider in the forest has woven a web. Long threads are attached to the branches. A butterfly has flown into this web. In one move, the butterfly or the spider can move along a thread segment to an adjacent intersection point; the butterfly can also escape to the end of a thread (branch) if it was previously in an adjacent intersection point. They take turns, starting with the butterfly. If the butterfly manages to reach the branches, it escapes (this is her victory). If the spider reaches the butterfly, it eats her (and this is his victory). It is also possible for the game to continue indefinitely without a winner.

![](https://cdn.mathpix.com/cropped/2024_05_06_26ec1b2817b0c07a544eg-28.jpg?height=451&width=458&top_left_y=2170&top_left_x=800)

a) How will the game end in the situation depicted in the image? (The web has four rings and seven radii.)

b) How will the game end if there are three rings and seven radii?

c) How will the game end if there are four rings and ten radii?

d) Analyze the general case: $K \geq 2$ rings and $R \geq 3$ radii.
g) Note that the butterfly always has a neutral strategy. It consists of the butterfly moving along its ring away from the spider. When $R>3$, it is obvious that the spider must move along its ring in the same direction, otherwise the butterfly will exit along the radius it is currently on. In this case, the position of the arthropods relative to the web and to each other does not change, so the butterfly can again make a similar move, and so on to infinity.

When $R=3$, the spider has another move - along its ring in the opposite direction, when it is on the same radius as the butterfly. But even then, the spider will not catch the butterfly. The butterfly moves in any direction along its ring, the spider - along its (it cannot move inward in the web - it will lose the butterfly), and so on to infinity. Consider the following butterfly strategy: move to the center of the web, and then leave along the radius farthest from the spider. In this case, the spider will not catch up through the center (it is at least one move behind), and it can intercept by moving along the ring (naturally, the outer one, the others are less advantageous) only if it has enough moves. This will be when $K \geq R / 2$ for even $R$ and when $K \geq (R-1) / 2$ for odd $R$.

Suppose this inequality is satisfied. If the butterfly moves along its ring, the spider pursues it, not allowing it to escape. As soon as the butterfly makes a move towards the center of the web, the spider stands on its radius. Then he maintains this property relative to her (for example, moving from the outer ring to the next and back), thereby not allowing her to escape until the butterfly reaches the center (here the spider needs to move to the outer ring, and if he is on it - move along it in any direction). Then the spider needs to see where the butterfly has moved and intercept it along the outer ring (in the direction where the path is shorter). Soon they will be on the same radius again (if the butterfly moves not only outward, it will lose time, and if it returns to the center again, the spider should act as described above). Then the spider's actions are repeated. Thus, in this case, neither has a winning strategy.

## Answer

a), b) Draw; c) the butterfly wins; g) when $K \geq [R / 2]$ draw, in other cases the butterfly wins.