Sam simplified the problem since I wrote this solution, making (1,0) be the point that everything else connects to. In the original problem there were just n equally spaced points with chords to one of them. Rotating the unit circle so that (1,0) was the point everything else connected to was the first insight that lead to a solution. Rotation of the unit circle is just the selection of a coordinate system, and does not change any geometric properties like chord lengths.

]]>This post by “Research in Practice” makes progress on the ellipse version of the problem, which I had dismissed as badly posed. He makes the problem explicit (scale the circle vertically after placing the points on the roots of unity), then comes up with a conjecture about the values, further generalizing the conjecture to other scaling factors, and making some fairly substantial progress on establishing the conjecture. It’s not done yet, but his post is worth reading.

]]>I did cut out one massive time-wasting side path I took, expanding as

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