The Unsolved Lollipop Problem - Numberphile

0:00 – 6:13

A Circle on a Stick Becomes a Puzzle

Picture a circle with a line jutting out perpendicular to its edge, like a pin through a balloon. Extend that line to infinity in both directions, and you have what mathematicians call a lollipop. One lollipop carves the plane into two regions: inside the circle and everywhere else. Now add a second lollipop. If you position them carelessly, you get seven regions (the interviewer catches this immediately when the mathematician undercounts). But arrange them with precision, overlapping the circles in a sliver and threading each stick through the other's circle, and suddenly you have 10 regions. The difference comes down to intersections: where circles cross circles, where sticks pierce circles, and where sticks meet sticks. Each new crossing births new regions. The formula is deceptively simple: regions equal intersections plus the number of lollipops plus one. With two lollipops crossing at seven points (two circle-circle, two stick-circle each way, one stick-stick), you get 7 + 2 + 1 = 10. The pattern holds for three lollipops too: arrange them so each pair crosses at seven points (21 total intersections), and you get 21 + 3 + 1 = 25 regions. The drawing looks like a tangle of surgical incisions, each stick carefully offset so no three lines meet at a single point, because triple intersections waste potential regions.