P10 · Packing and covering · Formed here

Equal circles in a plus sign

Place n circles of one common radius inside a plus sign (a 3 × 3 square with all four corner squares removed), making that radius as large as possible.

Formal definition

ContainerThe container is a square of side 3 with all four of its 1 × 1 corners removed, leaving a plus sign: the origin (0, 0) is its lower-left corner and (3, 3) its upper-right.
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

Our own variant: equal-circle packing in a plus sign (a 3 × 3 square with all four corner squares removed) was posed here, and there is no literature for it. Every n is unstudied; the standing record is all anybody knows.

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.