Place n circles of one common radius inside a half-disc of radius 1, making that radius as large as possible.
Formal definition
ContainerThe container is a half-disc of radius 1: its diameter lies along y = 0 from (0, 0) to (2, 0), centred at (1, 0), with the arc above.
SubmissionExactly n circles: one shared radius and n centres; every circle uses the same radius
ConstraintsEvery circle lies wholly inside the container; no two overlap in their interiors, tangency allowed
ObjectiveMake the common radius as large as possible
Getting a feel for it
Where the room for improvement is
Optimal packings are jammed contact structures: circles brace against each other and the boundary, with tilted rows, offsets, and the odd rattler touching nothing. Neat grids are almost never optimal.
Where the frontier is
Every best known value comes from Specht's csc survey table, which marks none of them proven; all fourteen n are open.