Figure-eight region-count conjecture

Prove or disprove the conjecture that the maximum number of regions formed by n figure-eight shapes is a_{\sf 8}(n) = 4n² − 3n + 2.

Background

A figure eight consists of two tangent circles of equal radius. Treating the two constituent circles separately gives an upper bound based on the arrangement of 2n circles, but the two circles within each figure eight cannot properly intersect.

The authors conjecture that this restriction reduces the attainable crossing count by exactly two per figure eight. The proposed formula agrees with computational or explicit constructions for n ≤ 3, but no general proof or construction is supplied.

References

We conjecture that the effect of this constraint is simply to reduce the number of crossings by 2 for each figure ${\sf 8}$, and so the solution for the ${\sf 8}$ is conjecturally that $a_{\sf 8}(n) = 4n2-3n+2$ (https://oeis.org/#1{A386486}).

— Cutting a Pancake with an Exotic Knife  (2511.15864 - Cutler et al., 19 Nov 2025) in Section 9.4