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P83 · Packing and covering · Classic · Hard

Equilateral triangles in a square · n = 10

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

Instancen = 10
ObjectiveMinimize container square side
Best known, unproven2.377Source-precision interval: 2.377–2.378 (values inside match)exhibited site certificate 2.377022803Compiled by Erich Friedman; Found by David W. Cantrell in July 2002. Source expression: s = 2.377+.

Formal definition

  • ContainerThe editor and certificate normalize to the unit square [0,1]². This is a coordinate convention only; scores use the literature's unit-piece scale. Boundary contact is allowed.
  • SubmissionSubmit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4.
  • ConstraintsExactly n congruent shapes lie wholly in the container with pairwise disjoint interiors. Polygon rotations are independent, not limited to quarter turns.
  • PrecisionInput decimals are exact rationals; regular-polygon vertices are algebraic, not rounded templates. Feasibility has no floating-point tolerance. A finite-decimal certificate is not a continuous optimality proof.
  • Objectivecontainer square side = 1/(2 sin(pi/3) r), smaller is better. Pages, leaderboards and literature share these units; conversion is automatic.
Open the full editor ↗
VERIFIED CONSTRUCTIONVERIFIED CONSTRUCTION

Getting a feel for it

Source and certificate

External targets are converted with their printed precision; decimal coordinate certificates are separate. Reconstruction loss never lowers the literature target, and a source construction is not automatically an optimality proof.

Source
The record-holding arrangement for Equilateral triangles in a square n = 10, 2.377022803
Current leader

2.377022803

container square side

Matches the best known
Answer sourceDavid W. Cantrell
Solution methodPublished reference construction
Challenge this record ↗Submit a proof / idea ↓Share a proof or idea in the discussion. Accepted contributions can earn proof points.
Record holder's solver noteNo solver note yet (expand)＋

The record holder has not shared a solver note yet.

ANSWER FORMAT

How to write your answer

the unit square [0,1]²

Submit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4.

The current leader's answer

{
  "placements": [
    {
      "turn": "0.267949192",
      "x": "0.500000001",
      "y": "0.878556009"
    },
    {
      "turn": "0.057686809",
      "x": "0.144826419",
      "y": "0.805013127"
    },
    {
      "turn": "0.502913674",
      "x": "0.855173582",
      "y": "0.805013126"
    },
    {
      "turn": "-0.502913672",
      "x": "0.289652837",
      "y": "0.610026253"
    },
    {
      "turn": "-0.057686807",
      "x": "0.710347164",
      "y": "0.610026252"
    },
    {
      "turn": "0.057686809",
      "x": "0.144826419",
      "y": "0.408040531"
    },
    {
      "turn": "0.502913674",
      "x": "0.855173582",
      "y": "0.408040530"
    },
    {
      "turn": "-0.267949192",
      "x": "0.289652836",
      "y": "0.121443991"
    },
    {
      "turn": "0.267949192",
      "x": "0.499999999",
      "y": "0.242887983"
    },
    {
      "turn": "-0.267949192",
      "x": "0.710347163",
      "y": "0.121443991"
    }
  ],
  "radius": "0.242887981"
}
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": "0.267949192",
      "x": "0.500000001",
      "y": "0.878556009"
    },
    {
      "turn": "0.057686809",
      "x": "0.144826419",
      "y": "0.805013127"
    },
    {
      "turn": "0.502913674",
      "x": "0.855173582",
      "y": "0.805013126"
    },
    {
      "turn": "-0.502913672",
      "x": "0.289652837",
      "y": "0.610026253"
    },
    {
      "turn": "-0.057686807",
      "x": "0.710347164",
      "y": "0.610026252"
    },
    {
      "turn": "0.057686809",
      "x": "0.144826419",
      "y": "0.408040531"
    },
    {
      "turn": "0.502913674",
      "x": "0.855173582",
      "y": "0.408040530"
    },
    {
      "turn": "-0.267949192",
      "x": "0.289652836",
      "y": "0.121443991"
    },
    {
      "turn": "0.267949192",
      "x": "0.499999999",
      "y": "0.242887983"
    },
    {
      "turn": "-0.267949192",
      "x": "0.710347163",
      "y": "0.121443991"
    }
  ],
  "radius": "0.242887981"
}

Submit {radius, placements:[{x,y,turn},…]}, with decimal strings of at most nine places. x,y are centres; circles require turn=0, polygons use turn=tan(θ/2). Coordinates and turn lie in ±4; 0 < radius ≤ 4. · Verifier v1.0.0

DISCUSSION

Discussion

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