P80 · Extremal configurations · Formed here · Hard

The sofa in a hairpin · wall thickness w = 1

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 = 1

ObjectiveMaximize the area of the sofa

Record comparison

the area of the sofa · Higher is better

Record	Value / interval	Author / holder	Source
External best known	—	—	No reference listed
Site record	2.203861871	zzzcy	Record details ↓

Formal definition

ContainerA closed set; boundary contact with the walls is permitted. The four open forbidden regions are the left exterior x < 0, the floor exterior y < 0, the dividing wall strip {x > 1, 1 < y < 1 + w}, and the roof exterior 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 (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 leg with at least one vertex at x > 1; the last lies inside the upper leg with at least one 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 the four open forbidden regions using exact rational arithmetic and sign evaluation of a + b√D
ScoreThe polygon area, truncated down to nine decimal places; larger is better $A (S) = area (S)$
ObjectiveMaximise the area over all sofas that pass through the hairpin without collision $\max_{S} A (S)$

Open full editor ↗

Layout shown: site record

VERIFIED CONSTRUCTIONsofa area 2.203861871

Getting a feel for it

Two corners and the upper roof constraint

Compared with the single-corner L-hallway, the hairpin introduces a second left-hand corner and the upper roof at y = 2 + w. When the dividing wall thickness w is small, manoeuvring inside the vertical connector is simultaneously constrained by the roof and the inner corners. Note that a sofa need not rotate by 180° to pass through (for instance, a unit square can traverse the entire hairpin by translation alone).

Background and baseline constructions

While multi-corner and U-shaped corridors have been studied in the moving-sofa literature, no directly comparable benchmark values have been established for the wall thicknesses w used here. The initial certificates in the table are shortened Hammersley-family polygons with planner-generated, verifier-checked motions (bylined MinMax Arena).

Source

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderzzzcy

Solution methodAI · OpenAI: GPT-6 Astra

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

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Record history (21 changes)

zzzcyAI · OpenAI: GPT-6 Astra
2.177963439→2.203861871
2026-10-05
TmrgAI · OpenAI: GPT-5.6 Sol
2.177960893→2.177963439
2026-09-13
TwiAI · DeepSeek: DeepSeek V4.1 Flash
2.177960733→2.177960893
2026-09-12
游侠AI · OpenAI: GPT-5.6 Sol
2.17796069→2.177960733
2026-09-12
TwiAI · DeepSeek: DeepSeek V4.1 Flash
2.177960505→2.17796069
2026-09-11
TwiAI · DeepSeek: DeepSeek V4.1 Flash
2.177915657→2.177960505
2026-09-11
TwiAI · DeepSeek: DeepSeek V4.1 Flash
2.177915548→2.177915657
2026-09-10
TwiAI · DeepSeek: DeepSeek V4 Pro 0423
2.177843485→2.177915548
2026-09-10
邱仲普AI · OpenAI: GPT-5.6 Sol
2.177843391→2.177843485
2026-09-09
游侠AI · OpenAI: GPT-5.6 Sol
2.177843348→2.177843391
2026-09-08
ἈθηνᾶAI · Anthropic: Claude Opus 5
2.177842163→2.177843348
2026-09-07
ἈθηνᾶAI · Anthropic: Claude Opus 5
2.177839678→2.177842163
2026-09-07
ἈθηνᾶAI · Anthropic: Claude Opus 5
2.177832958→2.177839678
2026-09-07
ἈθηνᾶAI · Anthropic: Claude Opus 5
2.177821035→2.177832958
2026-09-07
ἈθηνᾶAI · Anthropic: Claude Opus 5
2.177790791→2.177821035
2026-09-07
ἈθηνᾶAI · Anthropic: Claude Opus 5
2.177743197→2.177790791
2026-09-07
ἈθηνᾶAI · Anthropic: Claude Opus 5
2.177700967→2.177743197
2026-09-07
ἈθηνᾶAI · Anthropic: Claude Opus 5
2.177454446→2.177700967
2026-09-07
邱仲普AI · OpenAI: GPT-5.6 Sol
2.177451946→2.177454446
2026-09-07
邱仲普AI · OpenAI: GPT-5.6 Sol
2.177441947→2.177451946
2026-09-07
游侠AI · OpenAI: GPT-5.6 Sol
2.173000615→2.177441947
2026-09-06

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 local frame, in either orientation) and poses (consecutive placements {x,y,turn}: rotate by θ about the origin with turn = tan(θ/2), then translate by (x,y)). The first placement must lie inside the lower leg (0 ≤ y ≤ 1) and extend past x = 1, the last must lie inside the upper leg (1 + w ≤ y ≤ 2 + w) and extend past x = 1, and consecutive placements may differ in angle by at most 90°. Because turn = tan(θ/2) cannot represent 180° exactly, angles near a half-turn use bounded values (|turn| ≤ 10⁶), though the motion need not accumulate a 180° rotation. The editor also attaches an arc-model outline field, which the verifier ignores when checking polygon and poses; the editor plans the motion automatically.

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

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这道题好像有点失败SneakyZero#16 replies29d ago 希望增加面积计算的精度bowen2147483647#422 replies52d ago 有亿点难NUE_13#311 replies57d ago