P84 · Packing and covering · Classic · Hard

Equilateral triangles in a disc · n = 16

Pack n unit-side equilateral triangles into a disc. Contact is allowed; interiors cannot overlap. Minimise the container circle radius.

Instancen = 16

ObjectiveMinimize container circle radius

Record comparison

container circle radius · Lower is better

Record	Value / interval	Author / holder	Source
External best known	[1.687, 1.688)	Maurizio Morandi	Erich Friedman ↗
Site record	1.687452414	NUE_13	Record details ↓

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

Formal definition

ContainerThe editor and certificate normalise the container to the closed unit disc centred at (1,1), while public scores are reported in the literature's unit-piece scale. Boundary contact is allowed.
SubmissionSubmit {radius, placements:[{x,y,turn},…]}, with each value written as a decimal string of at most nine places. Here x, y are centre coordinates; circles require turn = 0 and polygons use turn = tan(θ/2). Coordinates and turn must lie in [−4, 4], with 0 < radius ≤ 4.
ConstraintsExactly n congruent pieces lie wholly inside the container with pairwise disjoint interiors. Polygons may rotate independently by arbitrary angles.
PrecisionInput decimals are parsed as exact rationals and regular-polygon vertices are defined algebraically, with zero floating-point tolerance in feasibility checks.
ObjectiveContainer circle radius = 1/(2 sin(π/3) r), where smaller is better. Pages, leaderboards, and literature records share this unit with automatic conversion.

Open full editor ↗

Layout shown: site record

VERIFIED CONSTRUCTIONVERIFIED CONSTRUCTION

Getting a feel for it

Literature targets and coordinate certificates

Literature targets are converted at their published precision, while nine-decimal coordinate certificates are stored and displayed alongside the source values.

Source

LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holderNUE_13

Solution methodAI · DeepSeek: DeepSeek V4.1 Flash

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 (1 changes)

NUE_13AI · DeepSeek: DeepSeek V4.1 Flash
1.687452419→1.687452414
2026-10-31

ANSWER FORMAT

How to write your answer

the closed unit disc centred at (1,1)

Submit {radius, placements:[{x,y,turn},…]}, with each value written as a decimal string of at most nine places. Here x, y are centre coordinates; circles require turn = 0 and polygons use turn = tan(θ/2). Coordinates and turn must lie in [−4, 4], with 0 < radius ≤ 4.

The current leader's answer

{
  "placements": [
    {
      "turn": "0.268220009",
      "x": "0.894658075",
      "y": "1.744835028"
    },
    {
      "turn": "-0.267678413",
      "x": "0.598439905",
      "y": "1.573613752"
    },
    {
      "turn": "-0.267678413",
      "x": "1.400980178",
      "y": "1.574019288"
    },
    {
      "turn": "0.268220009",
      "x": "0.598612795",
      "y": "1.231470653"
    },
    {
      "turn": "0.268220009",
      "x": "1.401153068",
      "y": "1.231876190"
    },
    {
      "turn": "-0.267678413",
      "x": "0.999877508",
      "y": "1.242407488"
    },
    {
      "turn": "-0.267678413",
      "x": "0.302394625",
      "y": "1.060249377"
    },
    {
      "turn": "-0.267678413",
      "x": "1.697544128",
      "y": "1.060954367"
    },
    {
      "turn": "0.268220009",
      "x": "1.000050398",
      "y": "0.900264390"
    },
    {
      "turn": "-0.267678413",
      "x": "1.296441458",
      "y": "0.729342568"
    },
    {
      "turn": "0.268220009",
      "x": "0.302567516",
      "y": "0.718106278"
    },
    {
      "turn": "0.268220009",
      "x": "1.697717019",
      "y": "0.718811269"
    },
    {
      "turn": "-0.267678413",
      "x": "0.703832228",
      "y": "0.729043113"
    },
    {
      "turn": "0.268220009",
      "x": "1.296614349",
      "y": "0.387199469"
    },
    {
      "turn": "0.268220009",
      "x": "0.704005119",
      "y": "0.386900015"
    },
    {
      "turn": "-0.267678413",
      "x": "1.000396179",
      "y": "0.215978193"
    }
  ],
  "radius": "0.342143141"
}

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

Instance parameters

{
  "n": 16,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.268220009",
      "x": "0.894658075",
      "y": "1.744835028"
    },
    {
      "turn": "-0.267678413",
      "x": "0.598439905",
      "y": "1.573613752"
    },
    {
      "turn": "-0.267678413",
      "x": "1.400980178",
      "y": "1.574019288"
    },
    {
      "turn": "0.268220009",
      "x": "0.598612795",
      "y": "1.231470653"
    },
    {
      "turn": "0.268220009",
      "x": "1.401153068",
      "y": "1.231876190"
    },
    {
      "turn": "-0.267678413",
      "x": "0.999877508",
      "y": "1.242407488"
    },
    {
      "turn": "-0.267678413",
      "x": "0.302394625",
      "y": "1.060249377"
    },
    {
      "turn": "-0.267678413",
      "x": "1.697544128",
      "y": "1.060954367"
    },
    {
      "turn": "0.268220009",
      "x": "1.000050398",
      "y": "0.900264390"
    },
    {
      "turn": "-0.267678413",
      "x": "1.296441458",
      "y": "0.729342568"
    },
    {
      "turn": "0.268220009",
      "x": "0.302567516",
      "y": "0.718106278"
    },
    {
      "turn": "0.268220009",
      "x": "1.697717019",
      "y": "0.718811269"
    },
    {
      "turn": "-0.267678413",
      "x": "0.703832228",
      "y": "0.729043113"
    },
    {
      "turn": "0.268220009",
      "x": "1.296614349",
      "y": "0.387199469"
    },
    {
      "turn": "0.268220009",
      "x": "0.704005119",
      "y": "0.386900015"
    },
    {
      "turn": "-0.267678413",
      "x": "1.000396179",
      "y": "0.215978193"
    }
  ],
  "radius": "0.342143141"
}

DISCUSSION

Discussion

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