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P85 · Packing and covering · Classic · Hard

Squares in an equilateral triangle · n = 12

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

Instancen = 12
ObjectiveMinimize container triangle side
Best known, unproven6.301Source-precision interval: 6.301–6.302 (values inside match)exhibited site certificate 6.301221383Compiled by Erich Friedman; Found by David W. Cantrell in July 2002. Source expression: s = 6.301+.

Formal definition

  • ContainerThe editor and certificate normalize to the equilateral triangle with vertices (0,0), (1,0), (1/2,√3/2). 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 triangle side = 1/(2 sin(pi/4) 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 Squares in an equilateral triangle n = 12, 6.301221383
Current leader

6.301221383

container triangle 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)＋

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ANSWER FORMAT

How to write your answer

the equilateral triangle with vertices (0,0), (1,0), (1/2,√3/2)

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": "0.131652498",
      "x": "0.483231436",
      "y": "0.678282006"
    },
    {
      "turn": "0.131652498",
      "x": "0.403881740",
      "y": "0.540844300"
    },
    {
      "turn": "0.131652498",
      "x": "0.574856573",
      "y": "0.519582613"
    },
    {
      "turn": "0.131652498",
      "x": "0.324532043",
      "y": "0.403406594"
    },
    {
      "turn": "0.131652498",
      "x": "0.495506877",
      "y": "0.382144907"
    },
    {
      "turn": "0.131652498",
      "x": "0.245182347",
      "y": "0.265968889"
    },
    {
      "turn": "0.414213562",
      "x": "0.670325773",
      "y": "0.238265021"
    },
    {
      "turn": "0.414213562",
      "x": "0.511626380",
      "y": "0.194401510"
    },
    {
      "turn": "-0.231047661",
      "x": "0.353251518",
      "y": "0.100843577"
    },
    {
      "turn": "0.414213562",
      "x": "0.171532847",
      "y": "0.080291971"
    },
    {
      "turn": "0.414213562",
      "x": "0.670325773",
      "y": "0.079392579"
    },
    {
      "turn": "0.414213562",
      "x": "0.829025166",
      "y": "0.079349696"
    }
  ],
  "radius": "0.112217416"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 12,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.131652498",
      "x": "0.483231436",
      "y": "0.678282006"
    },
    {
      "turn": "0.131652498",
      "x": "0.403881740",
      "y": "0.540844300"
    },
    {
      "turn": "0.131652498",
      "x": "0.574856573",
      "y": "0.519582613"
    },
    {
      "turn": "0.131652498",
      "x": "0.324532043",
      "y": "0.403406594"
    },
    {
      "turn": "0.131652498",
      "x": "0.495506877",
      "y": "0.382144907"
    },
    {
      "turn": "0.131652498",
      "x": "0.245182347",
      "y": "0.265968889"
    },
    {
      "turn": "0.414213562",
      "x": "0.670325773",
      "y": "0.238265021"
    },
    {
      "turn": "0.414213562",
      "x": "0.511626380",
      "y": "0.194401510"
    },
    {
      "turn": "-0.231047661",
      "x": "0.353251518",
      "y": "0.100843577"
    },
    {
      "turn": "0.414213562",
      "x": "0.171532847",
      "y": "0.080291971"
    },
    {
      "turn": "0.414213562",
      "x": "0.670325773",
      "y": "0.079392579"
    },
    {
      "turn": "0.414213562",
      "x": "0.829025166",
      "y": "0.079349696"
    }
  ],
  "radius": "0.112217416"
}

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

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