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P85 · Packing and covering · Classic · Hard

Squares in an equilateral triangle · n = 25

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

Instancen = 25
ObjectiveMinimize container triangle side
Best known, unproven8.464Source-precision interval: 8.464–8.465 (values inside match)exhibited site certificate 8.464101687Compiled by Erich Friedman; Found by Erich Friedman in 1997. Source expression: s = 5 + 2 √3 = 8.464+.

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 = 25, 8.464101687
Current leader

8.464101687

container triangle side

Matches the best known
Answer sourceErich Friedman
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 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": "0.514932285",
      "y": "0.301158731"
    },
    {
      "turn": "0.131652498",
      "x": "0.310297359",
      "y": "0.183012702"
    },
    {
      "turn": "0.131652498",
      "x": "0.114801060",
      "y": "0.080695239"
    },
    {
      "turn": "-0.131652498",
      "x": "0.512483597",
      "y": "0.726257150"
    },
    {
      "turn": "-0.131652498",
      "x": "0.571556612",
      "y": "0.623939687"
    },
    {
      "turn": "0.131652498",
      "x": "0.419304761",
      "y": "0.608111120"
    },
    {
      "turn": "-0.131652498",
      "x": "0.630629627",
      "y": "0.521622224"
    },
    {
      "turn": "0.131652498",
      "x": "0.487516403",
      "y": "0.489965091"
    },
    {
      "turn": "0.131652498",
      "x": "0.360231746",
      "y": "0.505793657"
    },
    {
      "turn": "-0.131652498",
      "x": "0.757914283",
      "y": "0.301158731"
    },
    {
      "turn": "-0.131652498",
      "x": "0.689702641",
      "y": "0.419304761"
    },
    {
      "turn": "0.131652498",
      "x": "0.428443388",
      "y": "0.387647628"
    },
    {
      "turn": "0.131652498",
      "x": "0.292020104",
      "y": "0.387647628"
    },
    {
      "turn": "-0.131652498",
      "x": "0.816987298",
      "y": "0.198841269"
    },
    {
      "turn": "-0.414213562",
      "x": "0.668080417",
      "y": "0.177219044"
    },
    {
      "turn": "0.131652498",
      "x": "0.455859270",
      "y": "0.198841269"
    },
    {
      "turn": "0.131652498",
      "x": "0.369370373",
      "y": "0.285330165"
    },
    {
      "turn": "0.131652498",
      "x": "0.232947090",
      "y": "0.285330165"
    },
    {
      "turn": "-0.414213562",
      "x": "0.872715343",
      "y": "0.059073015"
    },
    {
      "turn": "-0.414213562",
      "x": "0.754569314",
      "y": "0.059073015"
    },
    {
      "turn": "-0.414213562",
      "x": "0.636423284",
      "y": "0.059073015"
    },
    {
      "turn": "-0.414213562",
      "x": "0.518277254",
      "y": "0.059073015"
    },
    {
      "turn": "-0.414213562",
      "x": "0.400131225",
      "y": "0.059073015"
    },
    {
      "turn": "0.131652498",
      "x": "0.251224344",
      "y": "0.080695239"
    },
    {
      "turn": "0.131652498",
      "x": "0.173874075",
      "y": "0.183012702"
    }
  ],
  "radius": "0.083541858"
}
Submission format and technical detailsOpen this when you are ready to prepare a JSON answer＋

Instance parameters

{
  "n": 25,
  "sides": 5
}

The current leader's answer

{
  "placements": [
    {
      "turn": "0.131652498",
      "x": "0.514932285",
      "y": "0.301158731"
    },
    {
      "turn": "0.131652498",
      "x": "0.310297359",
      "y": "0.183012702"
    },
    {
      "turn": "0.131652498",
      "x": "0.114801060",
      "y": "0.080695239"
    },
    {
      "turn": "-0.131652498",
      "x": "0.512483597",
      "y": "0.726257150"
    },
    {
      "turn": "-0.131652498",
      "x": "0.571556612",
      "y": "0.623939687"
    },
    {
      "turn": "0.131652498",
      "x": "0.419304761",
      "y": "0.608111120"
    },
    {
      "turn": "-0.131652498",
      "x": "0.630629627",
      "y": "0.521622224"
    },
    {
      "turn": "0.131652498",
      "x": "0.487516403",
      "y": "0.489965091"
    },
    {
      "turn": "0.131652498",
      "x": "0.360231746",
      "y": "0.505793657"
    },
    {
      "turn": "-0.131652498",
      "x": "0.757914283",
      "y": "0.301158731"
    },
    {
      "turn": "-0.131652498",
      "x": "0.689702641",
      "y": "0.419304761"
    },
    {
      "turn": "0.131652498",
      "x": "0.428443388",
      "y": "0.387647628"
    },
    {
      "turn": "0.131652498",
      "x": "0.292020104",
      "y": "0.387647628"
    },
    {
      "turn": "-0.131652498",
      "x": "0.816987298",
      "y": "0.198841269"
    },
    {
      "turn": "-0.414213562",
      "x": "0.668080417",
      "y": "0.177219044"
    },
    {
      "turn": "0.131652498",
      "x": "0.455859270",
      "y": "0.198841269"
    },
    {
      "turn": "0.131652498",
      "x": "0.369370373",
      "y": "0.285330165"
    },
    {
      "turn": "0.131652498",
      "x": "0.232947090",
      "y": "0.285330165"
    },
    {
      "turn": "-0.414213562",
      "x": "0.872715343",
      "y": "0.059073015"
    },
    {
      "turn": "-0.414213562",
      "x": "0.754569314",
      "y": "0.059073015"
    },
    {
      "turn": "-0.414213562",
      "x": "0.636423284",
      "y": "0.059073015"
    },
    {
      "turn": "-0.414213562",
      "x": "0.518277254",
      "y": "0.059073015"
    },
    {
      "turn": "-0.414213562",
      "x": "0.400131225",
      "y": "0.059073015"
    },
    {
      "turn": "0.131652498",
      "x": "0.251224344",
      "y": "0.080695239"
    },
    {
      "turn": "0.131652498",
      "x": "0.173874075",
      "y": "0.183012702"
    }
  ],
  "radius": "0.083541858"
}

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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