We consider that two segments sharing an endpoint are not intersecting, and that endpoint is not counted as an intersection point. (This corresponds to reducing the chord length by \(eps\))

Consider an arbitrary black region formed when selecting an arbitrary set of chords.

When this region contains an arc, we assign a weight of \(1/2\) to each endpoint of the arc. When this region does not contain an arc, there is a unique vertex of this region that is farthest from the center of the circle. We assign a weight of \(1\) to this point.

Below, let’s consider the setting where we randomly select the set of chords to use.

Let’s focus on an arbitrary endpoint of an arbitrary chord and consider the expected value of the weight assigned here. First, there’s a probability of \(1/2\) that this chord is used. Then, regardless of how other chords are used, exactly one of the two regions near the endpoint we’re focusing on will be black, so this endpoint is assigned a weight of \(1/2\). Summarizing the above, we find that the expected value of the weight assigned to an endpoint is \(1/4\).

Next, let’s focus on the intersection point of two arbitrary chords and consider the expected value of the weight assigned here. First, there’s a probability of \(1/4\) that these two chords are used. Next, if there is no other chord that separates the center of the circle from this intersection point, the region corresponding to this intersection will always be white, so the weight is always \(0\). Conversely, if there is one or more such chords, a weight of \(1\) is assigned with probability \(1/2\). Summarizing the above:

• When there is no other chord separating the intersection from the center of the circle: expected value \(0\)
• When there exists another chord separating the intersection from the center of the circle: expected value \(1/8\)

Let \(I\) be the number of intersection points of two chords, and \(V\) be the number of \(i\) satisfying \(A_i + K \leq A_{i+1}\). Here, we set \(A_{N+1} = A_1 + L\).

The number of intersection points for which there is no other chord separating them from the center of the circle is \(N - V\).

Summarizing the above, the expected value of the total weight is \(\frac{3}{8}N + \frac{1}{8}I + \frac{1}{8}V\). Therefore, calculating this and multiplying by \(2^N\) gives the answer to the original problem.

Solution Example (C++)