P83 · Packing and covering · Classic · Hard

Equilateral triangles in a square · n = 17

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

Instancen = 17

ObjectiveMinimize container square side

Best known, unproven2.982Source-precision interval: 2.982–2.983 (values inside match)exhibited site certificate 2.982624084Compiled by Erich Friedman; Found by Maurizio Morandi in July 2008. Source expression: s = 2.982+.

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

Current leader

2.982624084

container square side

Matches the best known

Answer sourceMaurizio Morandi

Solution methodPublished reference construction

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

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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": "0.329449519",
      "y": "0.903214375"
    },
    {
      "turn": "0.267949192",
      "x": "0.664724760",
      "y": "0.903214375"
    },
    {
      "turn": "-0.267949192",
      "x": "0.497087139",
      "y": "0.806428750"
    },
    {
      "turn": "-0.267949192",
      "x": "0.832362380",
      "y": "0.806428750"
    },
    {
      "turn": "0.000000000",
      "x": "0.096785625",
      "y": "0.777353570"
    },
    {
      "turn": "-0.577350269",
      "x": "0.193571250",
      "y": "0.609715950"
    },
    {
      "turn": "0.267949192",
      "x": "0.474960855",
      "y": "0.612857499"
    },
    {
      "turn": "0.267949192",
      "x": "0.810236095",
      "y": "0.612857499"
    },
    {
      "turn": "-0.267949192",
      "x": "0.642598475",
      "y": "0.516071874"
    },
    {
      "turn": "-0.000000000",
      "x": "0.096785625",
      "y": "0.442078329"
    },
    {
      "turn": "-0.267949192",
      "x": "0.376079358",
      "y": "0.396982775"
    },
    {
      "turn": "-0.577350269",
      "x": "0.903214375",
      "y": "0.361208870"
    },
    {
      "turn": "0.267949192",
      "x": "0.335130804",
      "y": "0.203411524"
    },
    {
      "turn": "0.267949192",
      "x": "0.664724760",
      "y": "0.193571250"
    },
    {
      "turn": "-0.252278904",
      "x": "0.164724760",
      "y": "0.101664067"
    },
    {
      "turn": "-0.267949192",
      "x": "0.832362380",
      "y": "0.096785625"
    },
    {
      "turn": "-0.267949192",
      "x": "0.497087139",
      "y": "0.096785625"
    }
  ],
  "radius": "0.193571249"
}

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

Instance parameters

{
  "n": 17,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.267949192",
      "x": "0.329449519",
      "y": "0.903214375"
    },
    {
      "turn": "0.267949192",
      "x": "0.664724760",
      "y": "0.903214375"
    },
    {
      "turn": "-0.267949192",
      "x": "0.497087139",
      "y": "0.806428750"
    },
    {
      "turn": "-0.267949192",
      "x": "0.832362380",
      "y": "0.806428750"
    },
    {
      "turn": "0.000000000",
      "x": "0.096785625",
      "y": "0.777353570"
    },
    {
      "turn": "-0.577350269",
      "x": "0.193571250",
      "y": "0.609715950"
    },
    {
      "turn": "0.267949192",
      "x": "0.474960855",
      "y": "0.612857499"
    },
    {
      "turn": "0.267949192",
      "x": "0.810236095",
      "y": "0.612857499"
    },
    {
      "turn": "-0.267949192",
      "x": "0.642598475",
      "y": "0.516071874"
    },
    {
      "turn": "-0.000000000",
      "x": "0.096785625",
      "y": "0.442078329"
    },
    {
      "turn": "-0.267949192",
      "x": "0.376079358",
      "y": "0.396982775"
    },
    {
      "turn": "-0.577350269",
      "x": "0.903214375",
      "y": "0.361208870"
    },
    {
      "turn": "0.267949192",
      "x": "0.335130804",
      "y": "0.203411524"
    },
    {
      "turn": "0.267949192",
      "x": "0.664724760",
      "y": "0.193571250"
    },
    {
      "turn": "-0.252278904",
      "x": "0.164724760",
      "y": "0.101664067"
    },
    {
      "turn": "-0.267949192",
      "x": "0.832362380",
      "y": "0.096785625"
    },
    {
      "turn": "-0.267949192",
      "x": "0.497087139",
      "y": "0.096785625"
    }
  ],
  "radius": "0.193571249"
}

DISCUSSION

Discussion

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