Scale the rectangles 1×1, 1×2, …, 1×n by one common factor and fit them all into the unit square, making that factor as large as possible.
Formal definition
ContainerThe unit square: the origin (0, 0) at its lower-left corner, (1, 1) at its upper right
SubmissionSubmit placements in order 1×1 through 1×n, each a position plus an optional 90° rotation, together with the shared scale
ConstraintsAll rectangles use the one scale; rotation only by 90°; none overlap and none crosses the boundary
ObjectiveMake the shared scale as large as possible
Getting a feel for it
Where the room for improvement is
Strips from 1×1 to 1×n grow together, the narrow ones ever longer: upright or flat, whose seam hugs whose, a jigsaw where every piece must fit. Raise the scale a hair and all n strips must find new homes.
Where the frontier is
n ≤ 2 is trivial. The close relative — consecutive rectangles in similar rectangles — is collected on Friedman's Packing Center (the minrect page); this into-a-square variant has no per-n literature, and every other n is open.