MinMax Arena
中/EN
P84 · Packing and covering · Classic · Hard

Equilateral triangles in a disc · n = 10

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

Instancen = 10
ObjectiveMinimize container circle radius

Record comparison

container circle radius · Lower is better
RecordValue / intervalAuthor / holderSource
External best known[1.38468, 1.38469)Jonathan ViqueratErich Friedman ↗
Site record1.384681281condorRecord 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
The record-holding arrangement for Equilateral triangles in a disc n = 10, 1.384681281
LAYOUT & HISTORY

Record details

Inspect the layout, solver notes and record history.

Record holdercondor
Solution methodAI · OpenAI: GPT-6.1 Sol
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)
  1. condorAI · OpenAI: GPT-6.1 Sol
    1.384681284→1.384681281
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": "-2.549037667",
      "x": "0.452208062",
      "y": "0.762459278"
    },
    {
      "turn": "-0.797708994",
      "x": "0.839086627",
      "y": "1.705944657"
    },
    {
      "turn": "-1.748683048",
      "x": "1.367190773",
      "y": "1.624036785"
    },
    {
      "turn": "-2.378888449",
      "x": "1.181001479",
      "y": "0.948146846"
    },
    {
      "turn": "-0.797708986",
      "x": "0.757939625",
      "y": "1.045972905"
    },
    {
      "turn": "-2.378888447",
      "x": "1.194715549",
      "y": "0.302621460"
    },
    {
      "turn": "-2.549037669",
      "x": "0.533355058",
      "y": "1.422431029"
    },
    {
      "turn": "-0.582867860",
      "x": "1.572679965",
      "y": "1.261233866"
    },
    {
      "turn": "-0.759037161",
      "x": "0.790859070",
      "y": "0.406311911"
    },
    {
      "turn": "-0.759037162",
      "x": "1.486442354",
      "y": "0.600526205"
    }
  ],
  "radius": "0.416955351"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 10,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "-2.549037667",
      "x": "0.452208062",
      "y": "0.762459278"
    },
    {
      "turn": "-0.797708994",
      "x": "0.839086627",
      "y": "1.705944657"
    },
    {
      "turn": "-1.748683048",
      "x": "1.367190773",
      "y": "1.624036785"
    },
    {
      "turn": "-2.378888449",
      "x": "1.181001479",
      "y": "0.948146846"
    },
    {
      "turn": "-0.797708986",
      "x": "0.757939625",
      "y": "1.045972905"
    },
    {
      "turn": "-2.378888447",
      "x": "1.194715549",
      "y": "0.302621460"
    },
    {
      "turn": "-2.549037669",
      "x": "0.533355058",
      "y": "1.422431029"
    },
    {
      "turn": "-0.582867860",
      "x": "1.572679965",
      "y": "1.261233866"
    },
    {
      "turn": "-0.759037161",
      "x": "0.790859070",
      "y": "0.406311911"
    },
    {
      "turn": "-0.759037162",
      "x": "1.486442354",
      "y": "0.600526205"
    }
  ],
  "radius": "0.416955351"
}

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. · Verifier v1.0.0

DISCUSSION

Discussion

Discuss strategies, share construction methods, or ask questions. Any signed-in user can post under their public byline and unique #UID. New posts appear after moderation.

No threads yet. Start the first discussion.