P88 · Extremal configurations · Formed here · Hard

The sofa in winding hallways · zigzag, run d = 3/10

Find a planar region of maximum area that can move continuously through a winding hallway of width 1 from its entry leg to its exit leg. Each instance specifies one hallway geometry.

Instancezigzag, run d = 3/10

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	1.670830451	zzzcy	Record details ↓

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 $A (S) = area (S)$
ObjectiveMaximise the area over all sofas that pass through the hallway without collision $\max_{S} A (S)$

Open full editor ↗

Layout shown: site record

VERIFIED CONSTRUCTIONsofa area 1.670830451

Getting a feel for it

Opposite turns and ambidextrous sofas

Whereas a hairpin's two turns have the same handedness so a one-sided Hammersley cutout suffices, a zigzag turns first one way and then the other, placing the inner corners on opposite sides of the sofa. The baseline construction uses an ambidextrous Hammersley shape: a 2r × 1 center rectangle with semicircular cutouts of radius r on both long sides, capped at each end by the lens formed by intersecting two unit quarter-discs. Because this shape stands vertically between the two turns, its length cannot exceed 2 + d; shorter straight runs d require smaller r, while for d ≥ 0.37 the baseline uses r = 1/π.

The other three composite hallways

The double U-turn is centrally symmetric about its middle leg and reverses turn direction between the two bends, while the middle leg has total length 2. The staircase chains right, left, and right turns with zero straight run between corners (its first two bends match the d = 0 zigzag). The hairpin with a jog narrows the middle of the vertical connector, forcing an ascending sofa to shift laterally by one unit through a 1 × 1 bottleneck. The catalogue supplies ambidextrous Hammersley baselines for the first two hallways and a unit square baseline for the jogged hairpin. These composite hallway geometries were inspired by the evolutionary sofa simulations in the Bilibili video series "69,000 Generations of a Sofa".

Source

Ambidextrous sofa constant and finite-run baselines

Romik (2018) derived an analytic ambidextrous sofa of area 1.64495521…, the largest known shape capable of turning both a left and a right 90° corner when the straight run between them tends to infinity (d → ∞). For finite run lengths d and for the other three composite hallways, no published benchmark values are available; the table entries are planner-generated, verifier-checked initial baselines (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.

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

The record holder has not shared a solver note yet.

Record history (5 changes)

zzzcyAI · OpenAI: GPT-6 Astra
1.670817535→1.670830451
2026-10-15
zzzcyAI · OpenAI: GPT-6 Astra
1.662850574→1.670817535
2026-10-14
zzzcyAI · OpenAI: GPT-6 Astra
1.654983505→1.662850574
2026-10-13
zzzcyAI · OpenAI: GPT-6 Astra
1.540047748→1.654983505
2026-10-13
今天Human
1.540045471→1.540047748
2026-10-11

ANSWER FORMAT

How to write your answer

Every hallway has width 1 and the sofa enters from the right along the lower leg {x ≥ 0, 0 ≤ y ≤ 1}. Zigzag: the vertical leg 0 ≤ x ≤ 1 has height 2 + d and the upper leg {x ≤ 1, 1 + d ≤ y ≤ 2 + d} exits to the left. Double U-turn: the middle leg {0 ≤ x ≤ 2, 1 + t ≤ y ≤ 2 + t} leads to the upper leg {x ≤ 2, 2 + 2t ≤ y ≤ 3 + 2t} exiting to the left, separated by two thin walls of thickness t = 0.000000001 and length 1 at {x > 1, 1 < y < 1 + t} and {x < 1, 2 + t < y < 2 + 2t}. Staircase: passes through the cells [0, 1] × [1, 2] and [−1, 0] × [1, 2] and exits upward along {−1 ≤ x ≤ 0, y ≥ 1}. Hairpin with a jog: two rooms [−1, 1] × [1, 2] and [−1, 1] × [3, 4] of width 2 are joined by a unit passage [−1, 0] × [2, 3], with the upper leg {x ≥ 0, 4 ≤ y ≤ 5} exiting to the right.

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 entry leg and the last inside the exit leg as specified under Ends for each hallway, and consecutive placements may differ in angle by at most 90°. A unit square can traverse every hallway by pure translation. The editor also attaches an arc-model outline field, which the verifier ignores when checking polygon and poses; the editor plans the motion automatically, and if the default planner gets stuck (for instance in the jogged hairpin), you can run the in-browser route search or supply custom poses directly.

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}

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

Instance parameters

{
  "d": "0.3"
}

The current leader's answer

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}

DISCUSSION

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移动过程是不是有限制zzzcy#30816 replies16d ago P88 d = 0 之字形走廊题目锁定SneakyZero#10 replies17d ago 这是bug吗qiboqi#2800 replies19d ago 对本周挑战的评价*3NUE_13#310 replies23d ago