Optimal arrangements of lollipops

Determine the exact maximum number of regions formed by n lollipops, particularly whether the upper bound (7n² − 5n + 2)/2 remains attainable for n ≥ 4.

Background

A lollipop consists of a circle together with a ray emanating from its center, with the portion inside the circle removed. Two lollipops can intersect in at most seven points, yielding the upper bound R ≤ (7n² − 5n + 2)/2.

The bound is attained for n ≤ 3, but the authors report that the geometry of optimal arrangements becomes unclear from n = 4 onward. Numerical optimization is difficult because candidate arrangements contain very small regions and are sensitive to numerical errors.

References

This can be achieved for $n \le 3$ (Fig.\ \ref{FigLPOP3}), though it is already unclear what is happening when $n \ge 4$.

— Cutting a Pancake with an Exotic Knife  (2511.15864 - Cutler et al., 19 Nov 2025) in Section 9.6