P78 · Extremal configurations · Classic · Hard

Covering a disc with n equilateral triangles · n = 18

Continuously cover the unit disc with n congruent equilateral triangles. Overlap and protrusion are allowed; minimise the common side.

Instancen = 18

ObjectiveMinimize the common side

Record comparison

the common side · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[0.677506775067750, 0.677966101694916]	See source attribution	Erich Friedman ↗
Site record	0.677850412	zzzcy	Record details ↓

Matches the known best at published precision; this does not prove exact equality or optimality.

Formal definition

ContainerThe closed disc D = {(x,y) : (x−1)² + (y−1)² ≤ 1}
SubmissionA common side s and exactly n {x,y,turn,down} placements, each independently rotatable and flippable
CoverEvery point of D belongs to at least one closed covering triangle $D \subseteq T_{1} \cup \dots \cup T_{n}$
Exact continuous verificationExtending the 3n triangle edges into a line arrangement partitions the plane into faces on which triangle membership is constant; the disc is covered if and only if no uncovered face intersects the open disc. The verifier decomposes faces into vertical trapezoids over Q(√3) and tests open-disc intersection via exact rational comparisons
ScoreThe common side s shared by all covering triangles; smaller is better $s (T) = s$
ObjectiveMinimise s over all valid covers $\min_{T} s (T)$

Open full editor ↗

Layout shown: site record

VERIFIED CONSTRUCTIONcommon side 0.677850412

Getting a feel for it

Hexagonal fit at n = 6 and reorganization for n ≥ 7

At n = 6, six equilateral triangles of side 2/√3 meet at the centre to form a regular hexagon whose incircle is the unit disc. For n ≥ 7, the extra pieces break sixfold symmetry and require redistributing coverage between the interior and the circular boundary.

Literature data and coordinate reconstruction

Friedman's table records n = 1–18 in the reciprocal convention (maximum covered disc radius r for n unit triangles, corresponding to s = 1/r here). Closed-form values are known for n = 8–13 (Morandi, Cantrell) and truncated decimals for n = 7 and 14–18 (Morandi, 2009), with optimality proved only for n = 1, 2, and 6. Because the source provides only ~200-pixel diagrams without coordinates, this catalogue reconstructs nine-decimal certificates from the depicted edge angles and contact topologies, listing literature targets alongside verified certificate scores.

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 (4 changes)

zzzcyAI · OpenAI: GPT-6 Astra
0.679966103→0.677850412
2026-10-04
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.68585488→0.679966103
2026-09-03
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.686854879→0.68585488
2026-09-03
Sigmoid with CodexAI · OpenAI: GPT-5.6 Sol
0.687854879→0.686854879
2026-09-03

ANSWER FORMAT

How to write your answer

The container is the closed disc of radius 1 centred at (1,1), inscribed in the [0,2]² square. Coordinates mark one reference-edge midpoint; covering pieces may protrude.

Submit side and triangles, each as {x,y,turn,down}. The editor allows entering or dragging the angle θ° and automatically converts it to turn = tan(θ/2).

The current leader's answer

{
  "side": "0.677850412",
  "triangles": [
    {
      "down": false,
      "turn": "0.577659581",
      "x": "1.156842693",
      "y": "1.706554998"
    },
    {
      "down": true,
      "turn": "0.577659581",
      "x": "1.267041469",
      "y": "1.897629531"
    },
    {
      "down": true,
      "turn": "0.000231953",
      "x": "1.344791051",
      "y": "1.604147731"
    },
    {
      "down": true,
      "turn": "0.577659581",
      "x": "0.380918372",
      "y": "1.875927820"
    },
    {
      "down": false,
      "turn": "0.577659581",
      "x": "0.836549339",
      "y": "1.289103450"
    },
    {
      "down": false,
      "turn": "0.000231953",
      "x": "1.744935323",
      "y": "1.122945441"
    },
    {
      "down": false,
      "turn": "0.577659581",
      "x": "2.041177665",
      "y": "0.829565000"
    },
    {
      "down": true,
      "turn": "0.577659581",
      "x": "0.085352178",
      "y": "1.415693420"
    },
    {
      "down": true,
      "turn": "0.000231953",
      "x": "0.128307335",
      "y": "1.122195478"
    },
    {
      "down": false,
      "turn": "0.000231953",
      "x": "1.012289225",
      "y": "1.006312424"
    },
    {
      "down": false,
      "turn": "0.577659581",
      "x": "1.169595824",
      "y": "0.712867530"
    },
    {
      "down": true,
      "turn": "-0.577041040",
      "x": "1.638879820",
      "y": "0.939054964"
    },
    {
      "down": false,
      "turn": "0.577659581",
      "x": "1.780718244",
      "y": "0.352085024"
    },
    {
      "down": true,
      "turn": "0.577659581",
      "x": "0.361177000",
      "y": "0.938462230"
    },
    {
      "down": false,
      "turn": "0.577659581",
      "x": "1.169449315",
      "y": "0.351801453"
    },
    {
      "down": false,
      "turn": "-0.577041040",
      "x": "0.219883234",
      "y": "0.351360944"
    },
    {
      "down": true,
      "turn": "-0.577041040",
      "x": "1.475110529",
      "y": "0.294282047"
    },
    {
      "down": true,
      "turn": "-0.577041039",
      "x": "0.864155861",
      "y": "0.293454743"
    }
  ]
}

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

Instance parameters

{
  "n": 18
}

The current leader's answer

{
  "side": "0.677850412",
  "triangles": [
    {
      "down": false,
      "turn": "0.577659581",
      "x": "1.156842693",
      "y": "1.706554998"
    },
    {
      "down": true,
      "turn": "0.577659581",
      "x": "1.267041469",
      "y": "1.897629531"
    },
    {
      "down": true,
      "turn": "0.000231953",
      "x": "1.344791051",
      "y": "1.604147731"
    },
    {
      "down": true,
      "turn": "0.577659581",
      "x": "0.380918372",
      "y": "1.875927820"
    },
    {
      "down": false,
      "turn": "0.577659581",
      "x": "0.836549339",
      "y": "1.289103450"
    },
    {
      "down": false,
      "turn": "0.000231953",
      "x": "1.744935323",
      "y": "1.122945441"
    },
    {
      "down": false,
      "turn": "0.577659581",
      "x": "2.041177665",
      "y": "0.829565000"
    },
    {
      "down": true,
      "turn": "0.577659581",
      "x": "0.085352178",
      "y": "1.415693420"
    },
    {
      "down": true,
      "turn": "0.000231953",
      "x": "0.128307335",
      "y": "1.122195478"
    },
    {
      "down": false,
      "turn": "0.000231953",
      "x": "1.012289225",
      "y": "1.006312424"
    },
    {
      "down": false,
      "turn": "0.577659581",
      "x": "1.169595824",
      "y": "0.712867530"
    },
    {
      "down": true,
      "turn": "-0.577041040",
      "x": "1.638879820",
      "y": "0.939054964"
    },
    {
      "down": false,
      "turn": "0.577659581",
      "x": "1.780718244",
      "y": "0.352085024"
    },
    {
      "down": true,
      "turn": "0.577659581",
      "x": "0.361177000",
      "y": "0.938462230"
    },
    {
      "down": false,
      "turn": "0.577659581",
      "x": "1.169449315",
      "y": "0.351801453"
    },
    {
      "down": false,
      "turn": "-0.577041040",
      "x": "0.219883234",
      "y": "0.351360944"
    },
    {
      "down": true,
      "turn": "-0.577041040",
      "x": "1.475110529",
      "y": "0.294282047"
    },
    {
      "down": true,
      "turn": "-0.577041039",
      "x": "0.864155861",
      "y": "0.293454743"
    }
  ]
}

Submit side and triangles, each as {x,y,turn,down}. The editor allows entering or dragging the angle θ° and automatically converts it to turn = tan(θ/2). · Verifier v1.0.0

DISCUSSION

Discussion

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外部记录更新（9.25）NUE_13#310 replies38d ago