The sofa in winding hallways

Formal definition

• ContainerA closed set; boundary contact with the walls is permitted, while each wall interior is an open forbidden region. Zigzag: floor y < 0, left wall {x < 0, y < 1 + d}, inner corner quadrant {x > 1, y > 1}, and roof y > 2 + d. Double U-turn: floor y < 0, left wall {x < 0, y < 2 + 2t}, thin walls {x > 1, 1 < y < 1 + t} and {x < 1, 2 + t < y < 2 + 2t}, right wall {x > 2, y > 1}, and roof y > 3 + 2t, with t = 10⁻⁹. Staircase: floor y < 0, {x < 0, y < 1}, {x > 1, y > 1}, {x > 0, y > 2}, and {x < −1, y > 1}. Hairpin with a jog: floor y < 0, {x < 0, y < 1}, left wall x < −1, {x < 0, y > 4}, dividing wall {x > 1, 1 < y < 4} with lateral bulge {x > 0, 2 < y < 3}, and roof y > 5
• SubmissionA simple polygon (at most 800 vertices in the sofa's local frame) and a sequence of consecutive placements (at most 1500), with vertices × placements ≤ 300,000
• MotionBetween consecutive placements, the sofa moves by the unique rigid displacement connecting them: a pure translation when the turn values match, or (by Chasles' theorem) a single rotation of at most 90° about a fixed pivot
• EndsThe first placement lies inside the lower entry leg with at least one vertex past x = 1 (past x = 2 for the double U-turn). The last placement: in the zigzag, inside the upper leg with a vertex at x < 0; in the double U-turn, inside the upper leg (x ≤ 2) with a vertex at x < 0; in the staircase, inside {−1 ≤ x ≤ 0, y ≥ 1} with a vertex at y > 2; in the hairpin with a jog, inside the upper leg (x ≥ 0) with a vertex at x > 1
• Exact verificationEach placement's turn defines a rational rotation matrix and an exact pivot between consecutive placements. The region swept by each edge is represented by its exact boundary (a parallelogram for translations, or the two endpoint arcs, the inner envelope arc, and the initial and final segments for rotations) and checked against every open forbidden region using exact rational arithmetic and sign evaluation of a + b√D
• ScoreThe polygon area, truncated down to nine decimal places; larger is better$$A\left(S\right)=area\left(S\right)$$
• ObjectiveMaximise the area over all sofas that pass through the hallway without collision$$\underset{S}{\max }A\left(S\right)$$