Place n freely rotatable squares of one common side inside a circle of radius 1, none overlapping, making that side as large as possible.
Formal definition
ContainerA circle of radius 1 centred at (1, 1), so both coordinates run from 0 to 2
SubmissionEach square is {cx, cy, ux, uy}: a centre plus one half-edge vector, the other fixed as (−uy, ux)
ConstraintsEvery square tilts freely; all lie inside the container; no two overlap in their interiors, touching allowed. Only the smallest square is scored, so unequal sides gain nothing
ObjectiveMake the common side as large as possible; compared internally by its square, exactly
Getting a feel for it
Where the room for improvement is
A circular container has no corners: straight sides cannot hug the arc, so optima are almost always tilted, squares bracing corner against corner.
Where the frontier is
n = 4 is proven; the other best known values come from Friedman's squares-in-circles survey (many contributors since 1997), none of them proven.