hackercupai/hackercup huggingface dataset:
Let's call a position where the circle doesn't fall *stable*. First, let's see how we can check whether a given point is stable. Consider all intersection points between that cup and surrounding positions (we can find these naively in linear time). If all of these points lie on a half-circle, then the ring will fall. Otherwise, there will be three points that create a triangle which contains the center of mass of the ring.

**Solution 1:**

As it turns out, any stable point in general form (we can ignore points that are edge cases because those points are ~\(0\%\) of the area) will have at least \(3\) intersections. In fact, if a point is stable it's true that there are a particular \(3\) intersection points that make it stable. This idea is informally supported with what we'll call the "Jenga Principle": if a ring is balancing on \(4\) or more points, one of them will be "loose" in the same way a Jenga block resting on three blocks below it will always be resting on at least one "loose" block which could be removed.

We can case out all of the possible ways that two or three line segments could support a ring, brute force those segments, and then compute the overlap of these shapes. This is tricky to do mathematically for the area on which a ring could balance on two segments (the shapes include more than just circles and line segments), so some method of curve approximation will be useful here.

**Solution 2:**

Actually, if we were to just pick random points \(10^6\) times, we'll get a reasonable approximation of the answer. To get even closer, there are other tricks we can try: first, we can better distribute our points to decrease variance, such as overlaying them evenly in a grid. Once we're there, there are several options to improve our sampling, and we can follow one or more to get a solution which is accurate enough and runs in time:

- Spend less time looking at spaces that are surrounded by points that are either all stable or all unstable: Because the bounds are small, these points are very unlikely to differ from their neighbors once the neighbors are close enough.
- Use a *k*-d tree or similar data structure to break up the search space, and stop searching if large areas are \(100\%\) similar after many samples.
- Multithread our solution and optimize the constant factor overhead.
- Analyze the variance of the solution ahead of time to determine how many samples are needed for a sufficiently accurate estimate.

This optimization can be a tedious but rather straightforward process, and the main challenge is not underestimating the complexity of the potential solutions, or spending too much time on implementation. Notably, the stable region isn't necessarily convex, isn't necessarily entirely located within the polygon, isn't necessarily connected, and can even be connected but with holes.