What is the maximum number of pairwise intersections between a square and two ellipses?
Two ellipses can intersect in at most four points, but that's all I have within me…
2026-04-25 07:43:43.1777103023
Maximal pairwise intersections between a square and two ellipses
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Note that two ellipses can cross in at most four points, and that a line segment can cross an ellipse in at most two points. It follows that the maximal number of pairwise intersections between two ellipses and a square is $2\cdot4\cdot2+4=20$.
Here is an arrangement that achieves that maximum.