P80 · Extremal configurations · Formed here · Hard

The sofa in a hairpin · wall thickness w = 5/4

Find the planar region of largest area that can be moved continuously through a hairpin hallway of width 1: two right-angled corners the same way round, with a wall of thickness w between the legs.

Instancewall thickness w = 5/4

ObjectiveMaximize the area of the sofa

Formal definition

ContainerA closed set; touching a wall is allowed. The forbidden regions are four open sets: the left wall x < 0, the floor y < 0, the wall strip {x > 1, 1 < y < 1 + w} and the roof y > 2 + w $H = {x \geq 0, 0 \leq y \leq 1} \cup {0 \leq x \leq 1, 0 \leq y \leq 2 + w} \cup {x \geq 0, 1 + w \leq y \leq 2 + w}$
SubmissionA simple polygon (its vertices in the sofa's own frame, at most 800) and its consecutive placements (at most 1500), with vertices × placements at most 300,000
MotionBetween consecutive placements the sofa moves by the unique rigid displacement taking one to the other: a translation when the turns agree, otherwise (Chasles) one rotation about a fixed pivot, by at most 90°
EndsThe first placement lies inside the lower leg with a vertex at x > 1; the last lies inside the upper leg with a vertex at x > 1
Exact verificationEach placement's turn gives a rational rotation matrix and the pivot between placements is solved exactly; the region swept by each edge is compared with the four open forbidden regions through its boundary — a parallelogram for a translation, the two endpoint arcs, the inner arc and the first and last segments for a rotation. All arithmetic is rational; the arcs enter only through the sign of a + b√D. No sampling and no epsilon
ScoreThe area of the polygon, cut down to nine decimals; larger is better $A (S) = area (S)$
ObjectiveMaximise the area over every sofa that gets through the hairpin $\max_{S} A (S)$

Open the full editor ↗

VERIFIED CONSTRUCTIONsofa area 2.205991826

Getting a feel for it

Why this is not the L

The hairpin adds a roof and a second corner, so a motion valid in the L may collide here. The supplied baseline uses a shortened Hammersley-family sofa along a planned two-corner path that must clear the roof. This explains that construction, not an optimality bound for all shapes and motions. A half-turn is not required: a unit square can pass by translation alone.

Where the frontier is

U-shaped and other multi-corner corridors have appeared in prior work, but the site has not verified directly comparable external best values for these wall thicknesses. The starting constructions are shortened Hammersley-family shapes with planner-generated, verifier-checked motions, bylined MinMax Arena. They are baselines, not known optima.

Source

Current leader

2.205991826

the area of the sofa

Answer sourceMinMax Arena

Solution methodMinMax Arena reference construction

Challenge this record ↗Submit a proof / idea ↓Share a proof or idea in the discussion. Accepted contributions can earn proof points.

Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

ANSWER FORMAT

How to write your answer

The hallway is H = {x ≥ 0, 0 ≤ y ≤ 1} ∪ {0 ≤ x ≤ 1, 0 ≤ y ≤ 2 + w} ∪ {x ≥ 0, 1 + w ≤ y ≤ 2 + w}: the lower leg comes in from the right, the vertical leg is on the left, the upper leg goes back out to the right. Between the horizontal legs is the wall {x > 1, 1 < y < 1 + w}; above everything is the outer wall y > 2 + w.

Submit polygon (the vertices in the sofa's own frame, either orientation) and poses (consecutive placements {x,y,turn}: turn by θ about the origin with turn = tan(θ/2), then shift by (x,y)). The first placement must lie inside the lower leg (0 ≤ y ≤ 1) reaching past x = 1, the last inside the upper leg (1 + w ≤ y ≤ 2 + w) reaching past x = 1, and consecutive placements may differ by at most 90°. The turn parameter cannot represent exactly 180°; a finite turn may approach it (|turn| ≤ 10⁶). A motion need not accumulate a half-turn. The editor also attaches outline (its arc model); the verifier ignores it and judges polygon and poses alone. In the editor the motion is planned automatically for the drawn shape.

The current leader's answer

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Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
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The current leader's answer

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}

DISCUSSION

Discussion

Talk strategy, share methods, ask why you are stuck. Every signed-in user can post. Posts carry your public byline, the same name your records use; the #number after it is the account's signup ordinal, so a name cannot be worn by someone else. New posts appear after an automated review.

有亿点难NUE_13#310 replies1h ago