Place n non-overlapping equal circles inside a 2 × 1 rectangle, making the common radius as large as possible.
Formal definition
ContainerA rectangle 2 wide and 1 tall: the origin (0, 0) at its lower-left corner, (2, 1) at its upper right
SubmissionExactly n circles: one shared radius and n centres
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
n = 3..6 are proven by elementary zig-zag arguments (see each sub-problem's known-best row); larger n in the 2 × 1 rectangle have no systematic table and are all open.
ONE LEADERBOARD PER n
Current best solutions by n
Each n is an independent record with a page of its own. Open any of them to inspect the current construction, then challenge it.